text:
Andreas Kolitsch
Haus am Schlossbach
Winzererstr. 21
83098 Brannenburg
Gewerbeamt Stadt Brannenburg
83098 Brannenbur
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info@hausamschlossbach.de
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Alle Aufnahmen wurden selbst von Mitgliedern und Freunden der Familie Kolitsch getätigt.
Stand der AGB Sept.2018
Gratis AGB erstellt von agb.de


Situated in the Massinga District in Inhambane Province of Mozambique on a 100 hectare estate and 600 kms north of Maputo, 110 kms north of Inhambane and 170 kms south of Vilanculos.
Rates are quoted in USD and paid in ZAR, equivalent to the USD on the day payment is made. Rates are subject to change due to USD Fluctuation. SA/SADC Residents Rand Rates are available upon request.
Tea, coffee (excluding speciality tea/coffee), bottled water, fruit juice, local beer, local soft drinks and house wine (house wine with meals only).
A Beach Bar and Pool nestled right at the edge to the warm sea provide comfort and drinks to sun lovers. Nearby the boulders that have been artistically chiseled by the elements, provide a unique feature and shade for secluded picnics. Comfort is a priority in one of the 16 Ocean Front Deluxe rooms that were designed for Romantics. The 12 Ocean View Chalets, also with unimpeded views of the sea, are designed with families in mind. Regrettably the resort does not accommodate kids under 6 years.
In season, you can even view whales frolicking from your own deck whilst sipping exotic cocktails. Later, the night sky lights up and you feel like reaching out to catch the falling stars. Massinga Beach offers guests the opportunity to experience the splendor of Mozambique in a truly luxurious setting.
Massinga Beach is accessible by flights to Inhambane from where the lodge provides airport transfers on request. Self-drive is possible on a good paved road, heading north from Maputo to Massinga town. Last 10kms a 4×4 vehicle is required, however, parking facilities are available in Massinga Town for 2×4 vehicles.
Facilities include a Leisure Centre with a 180° view of the coastline and Indian Ocean beyond plus a Beach Bar and swimming pool right at the edge of the warm sea.
Various beach games, kayaking, table tennis and pool table at the beach bar plus snorkelling and rock-fishing a mere +-2km walk up the pristine beach (arrangements may be made for vehicle transfer) and disc golf in the palm groves are available.
In-room massage therapy, seasonal whale-watching and amazing star-gazing all add to Massinga Beach's appeal.
Spacious en-suite comprising of a bedroom with a 180 degree ocean view, front deck with private splash pool & sun loungers, extra length king-size bed, luxury linen, wrap around mosquito net curtain, bathrobe, room safe, stocked mini bar, tea & coffee station & air conditioning. En-suite bathroom with direct access to deck's freestanding bath, rain shower, seperate toilet and luxury guest amenities.
Spacious Suite. Bedroom with a 180 deg ocean view, extra length king-size bed, luxury linen, wrap around mosquito net curtain, room safe, air-conditioning in the main bedroom. En suite bathroom has a freestanding bath, rain shower, seperate toilet and luxury guest amenities. Lounge with double sleeper couch and seperate shower/basin/toilet. Certain rooms are configured with 2 single beds in the lounge area. Stocked mini bar and tea & coffee station. Bedroom and lounge lead onto deck with outside patio furniture.
Double or twin beds with en-suite bathroom. Mosquito net curtain, air conditioning/fan, mini bar and tea & coffee station. All rooms lead onto a deck with outside patio furniture. Suitable for guides and tour leaders.
Enquire about our bush and beach combo (3 nights at Lukimbi Safari Lodge and 4 nights at Massinga Beach) – great savings of R3000.00 per person per stay!
Massinga Beach is accessible by flights to Vilanculos and Inhambane. The lodge provides airport transfers whilst self-drive is possible on a good paved road from Maputo to Massinga. The last 10kms is on a scenic sand road and suitable for 4×4 vehicles. Parking is available upon request in Massinga Town at a nominal daily rate.
Meet and Greet at Inhambane Airport. A short drive through the historic town of Inhambane to the port to catch a private local motorized ferry to Maxixe (+/- 25 min). This will give you a great insight into the daily life of the local commuter as well as a photographic opportunity. From Maxixe to the Massinga Beach turnoff is a +- 45 min road transfer along the tarred EN1 main road, passing through various towns and villages and crossing the Topic of Capricorn. A 10km sand road through the coconut groves will bring you to the splendor of Massinga Beach. Complimentary water and soft drinks are provided during the road transfer. Duration, app 2 hours. Transfer without the local motorized ferry option also available on request.
Meet and Greet at Vilanculos Airport. A 170km road transfer passing villages and spectacular Boabab trees will bring you to the Massinga Beach turnoff. Complimentary water and soft drinks are provided during the road transfer. Duration, appr 2.5 hours.
Snorkeling at "Fingers" Rock Formation, Rock and Surf Fishing, Beach Games & Bicycle Riding, Massage Therapy, coconut Cutting Demonstration, Nature walks both on the beach and through the hillside, Board Games, Pool Tables, Table Tennis, Darts, Satellite Television, Star Gazing – telescope provided on the leisure deck. Beach Picnics seasonal and by prior arrangement.
All reservations that have not been confirmed with a 30% Non-refundable deposit are deemed as Provisional and may be released at the Property's discretion, with or without notice. During high season we require a 50% non refundable deposit. Balance of payment is due 40 days prior to arrival. For bookings made within 40 days of travel, full payment is required on confirmation. For bookings cancelled 45 - 31 days prior to arrival: 30% cancellation fee For bookings cancelled 30 - 0 days: 100% cancellation fee Refunds , if agreed, are subject to admin fees.
Regret no children under 6 years of age are permitted. Children 6 to 12 years are accommodated with parents in an Ocean View Chalets at Child Rate.
Massinga Beach sits above a stretch of endless, remarkably private beach overlooking the azure Indian Ocean. Far from the hustle and bustle of city life and offering barefoot luxury, the lodge caters for honeymooners and romantics.


Would you spend considerable time considering your teeth? You ought to, given that men and women recognize your tooth more than it may seem. If your dental care video game needs suggestions, this post will enable you to. We've put together a listing of dental care tips to provide you with the smartest laugh at any time.
Use only mouth wash free from liquor. Liquor for an element can dry up the muscle tissues in your jaws. This generates a reproduction terrain for the really bacteria you are hoping to avoid. Alcohol-free mouth wash can also be less of a urge for youngsters and teens looking to try when bored.
Floss, floss, floss! Certain anyone brushes their teeth like they're expected to, but how many of them actually floss. Flossing washes the region between teeth exactly where meals could get caught up and microorganisms can distributed. This may also stop the start of probable infection that could happen when you don't floss.
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If you buy tooth paste, you need a single with fluoride whether it be organic or otherwise not. This may build tooth energy. Strong teeth eventually are healthier kinds.
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You shouldn't merely remember to brush the surface areas of the tooth. It's important to clean the gum line also, receiving beneath them to clear out anything at all which is placed there. Oral cavaties that are beneath the gum line will be the worst as a result, you must do all you are able to clean this area.
Do you know that the sequence that you take in your meals in can be a factor in regardless of whether you produce cavities? Having sugary meals including food by itself is much worse for the the teeth, than when you eat this cake right after a dinner. Don't eat sugary foods by yourself, generally try to eat them with other food items.
In order to avoid critical and achievable long-lasting harm to your the teeth, never ever use them for any action apart from nibbling the meal that you just take in. You have the chance of cracking or cracking your pearly whites if you utilize your tooth to open up a deal, draw something which is stuck or fracture wide open nutshells.
There are numerous strategies to much better your teeth's well being, but one is extremely important. That may be maintaining semi-twelve-monthly sessions along with your dental office. These visits are used to clear your teeth and perform check ups. If fragile areas can be found with your the teeth, your dentist may possibly fill up these places to offer your the teeth a lot more protection.
After learning more about ways to care for your teeth, you should be able to put into action a number of modifications. Let the recommendations become the perfect information, and give oneself the look you usually needed. Maintain discovering what you can, as well as your pearly whites will appear better over the years.


2014_Pulitzer_Becherer_002: Global News Events: Global News Portfolio: New Orleans based Photojournalist covering the Deep South and beyond.
GAZA CITY, GAZA STRIP. Explosions caused by Israeli airstrikes erupt in the Jabal al-Rayyes neighborhood of Gaza City in the Gaza Strip on July 29, 2014. Israel said they were targeting 32 tunnels dug to allow Hamas militants to go under the border fence and infiltrate Israel.


Scoring -- Florence: D. Spavins 16; AOSR: Mengxiong Jiang 12, Ahmed Assad 10. Noteworthy -- Andrea Pettanello grabbed 10 rebounds for AOSR, while Assad added 8.
Scoring -- Florence: Spavins 9; AOSR: Assad 16, Cyrus Watkins 12. Noteworthy -- Assad and Pettenello each had 10 rebounds for AOSR; Watkins added 5 assists.
Scoring - Brussels: Jalen Pollard 31, Kalani Villa 14, Zafer Ozkeroglu 3; Alconbury: Derick Caples 13, Will Besterman 13, Spencer Swapp 11. Noteworthy -- Liam Prendergast led Alconbury with 9 rebounds, while Swapp and Elijah Campbell each had 8 rebounds.
Scoring - Spangdahlem: Montgomery 15, Rouse 13, E. Dejesus 9; Sigonella: Sean Solano 10, Greg Neal 8, Johnathan English 6.
Scoring -- SHAPE: George Vlachakis 8, Nathaniel Barden 6; Vilseck: Chris Graveley-Rosado 15, Xxavier Bynes 11. Noteworthy -- Bynes made 5 steals, while Josh Valembrun grabbed 11 rebounds for Vilseck.
Scoring -- BFA: Jacob Fortune 26; Wiesbaden: Markez Middlebrooks 14, Roberto Eiseman 13, Tyrese Harris 11, Dom Bivins 10. Noteworthy -- Eiseman had 10 rebounds in a double-double performance.
Scoring -- Florence: Kim Huijsmans 12; AOSR: Amaya Vaquerizo 15. Noteworthy -- Huijsmans had 11 rebounds, while Fleur Huijsmans grabbed 8 for Florence; Sophie Barber (6) and Henna Moussavi (5) paced AOSR on the glass.
Scoring -- Florence: K. Huijsmans 11, Sophia Lovalvo 6; AOSR: Vaquerizo 18, Isabella Walsh 12. Noteworthy -- K. Huijsman grabbed 9 rebounds for Florence; Claudia Ricci handled 5 rebounds for AOSR.
Scoring -- Alconbury: Thorne 19, Palacios 9; Brussels: Nina Duvnjak 7, HannahHite 6. Noteworthy -- Friedhoff had 12 rebounds for Brussels, while Duvnjak dished out 4 assists.
1, Zoe Kinzie (Vil) 291; 2, Trinity Lepine (Stu) 287; 3, Karla Redolosa (Stu) 286; 4, Angelisa Dobbins-Taylor (Vil) 284; 5, Eve Walter (Stu) 283.


text:
\section{Introduction}

The analysis of the gravitational thermodynamics  has suggested
that the entropy of a large number of $N$ coincident branes should obey
a power law scaling $N^2, N^{3/2}, N^3$ for D-branes, M2-branes and
M5-branes respectively \cite{KT}. For coincident D-branes, the $U(1)$ gauge
symmetry for individual D-brane is enhanced to a $U(N)$ gauge symmetry,
and the $N^2$ dependence of the entropy is nicely accounted for by the
$N^2$ degrees of freedom present in the supersymmetric $U(N)$ Yang-Mills
description of the coincident D-branes. The situation is much less clear
for multiple M2 or M5-branes. Some of the outstanding questions are,
for example: How does the gauge symmetry get enhanced for coincident M-branes?
What is the appropriate mathematical description?

Recently a new class of (2+1)-dimensional superconformal field theories
with maximal $\cN=8$ supersymmetry has been constructed
by Bagger and Lambert \cite{BL1,BL2,BL3},
and separately by Gustavsson \cite{G1}. These field theories have
been proposed as the low energy effective field theories for
coincident M2-branes and thus  provide a nonperturbative description of
M-theory on $AdS_4 \times S^7$ according to the AdS/CFT correspondence
\cite{adscft}.
Another proposal is due to Aharony, Bergman, Jafferis and Maldacena
\cite{ABJM} which proposed a certain $\cN=6$ superconformal
Chern-Simons-matter theory as  the theory describing the
worldvolume of multiple M2-branes at low energies \cite{adscft-ABJM}.
In this description,
a total of $\cN=6$ supersymmetry is explicitly realized \cite{susy}.

One of the most exciting features of the BLG theory is
the use of a new mathematical
structure, a metric Lie 3-algebra, in its description of the gauge symmetries
of the multiple M2-branes. It has also been suggested that
Lie 3-algebras also play a crucial role in the description of multiple M5-branes
 \cite{CS1,CS2,LP}.

In the case of D-branes, the worldvolume theory
carries a gauge symmetry that
is described by a semisimple Lie algebra. Indeed
the class of semisimple Lie algebras is
distinguished in the theory of Lie algebras.
Physically,
semisimple Lie algebras are the Lie algebras of compact connected Lie
groups, one that are universally used to describe the continuous
symmetries of physical systems.
Mathematically, semisimple Lie
algebras are completely classified and are fully understood.
In particular,
semisimple Lie algebras admit an elegant and very simple
basis of generators called the {\it Cartan-Weyl basis}. See
equations \eq{metric-cA} and \eq{cw2-1} below.
Motivated by the simplicity of the structure of a
Cartan-Weyl basis of generators for a semisimple Lie algebra,
one can introduce the notion
of a Cartan-Weyl basis for a metric Lie 3-algebra. A Cartan-Weyl basis
consists of the generators
${H_I, E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}}$ where $H_I$ ($I =1, \cdots, N$ for some
$N$) form a Cartan subalgebra $\cH$ of
the Lie 3-algebra and $E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}$ are labelled by {\it roots} $\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}$, which are
linear functions on $\cH^{\wedge 2}$.
The generators  satisfy the Lie 3-brackets
\begin{equation} \label{ccw1}
[H_I,H_J,H_K]= 0
\end{equation}
and
\begin{align} \label{ccw2}
\; [H_I, H_J, E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}]& =& \alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}_{IJ} E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}, \nonumber\\
\; [H_I,E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}, E^\b] &=&  \begin{cases}
\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}_{IK} g^{KL} H_L, & \mbox{if $\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}+ \b =0$}, \\
g_I(\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A},\b) E^{\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}+\b}, & \mbox{if $\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}+ \b \neq 0$ is a root},\\
0,  & \mbox{if $\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}+ \b$ is not a root},
\end{cases}
\\
\; [E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A},E^\b, E^\g] &=&  \begin{cases}
-g_K(\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A},\b) g^{KL} H_L,  & \mbox{if $\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}+ \b + \g =0$}, \\
c(\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A},\b,\g) E^{\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}+\b+\g},  & \mbox{if $\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}+ \b + \g \neq 0$ a root},\\
0,  & \mbox{if $\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}+ \b + \g$ is not a root};
\end{cases}
\nonumber
\end{align}
and they are normalized such that
\begin{equation}
\langle E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}, E^\b \rangle = \d^{\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}+\b}, \quad \langle H_I, E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A} \rangle =0, \quad
\langle H_I, H_J \rangle  = g_{IJ}.
\end{equation}
Here $\langle \cdot , \cdot\rangle$ is the metric and the part
$g_{IJ}$ is assumed to be invertible  with the inverse $g^{IJ}$.
We will call a metric
Lie 3-algebra a {\it Cartan-Weyl 3-algebra}
if it admits such a {\it Cartan-Weyl basis}.
It is natural to speculate that a
classification theorem may be obtained for the
Cartan-Weyl 3-algebras.

Due to their similarity in structure to semisimple Lie algebras,
it is  natural to speculate  that Cartan-Weyl 3-algebras
may also play a role in the description of certain yet to be discovered
generalized symmetries of Nature.
The understanding of the structure and the classification of Cartan-Weyl
3-algebras are therefore potentially important.
To achieve these goals is the main motivation of this paper.
Once this is achieved, it is natural to try to see whether and how
this kind of Lie 3-algebras is useful for the BLG theory. This is another
motivation of this work.

The organisation of the paper is as follows.
In section 2.1, we review some basic facts about Lie algebras. In
particular we recall the definition of a Cartan-Weyl basis.
In section 2.2, we  motivate and introduce the definition of
a Cartan-Weyl basis for a
metric Lie 3-algebra. The consistency conditions arising from the fundamental
identity are analysed in details in the appendix.
The resulting conditions that the two-form roots $\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}_{IJ}$ and
the coefficients $g_I(\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A},\b)$
and $c(\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A},\b,\g)$  have to satisfy are
summarized in section 2.3. In section 3, these conditions are  solved
fully and a complete classification of Cartan-Weyl 3-algebras is obtained.
In section 4, we
consider the embedding of $\cA_4$
\footnote{The Lie $n$-algebra
$\cA_{p,q}$ is a metric Lie $n$-algebra with signature $(p,q)$, $p+q
=n+1$.
It has $n+1$ generators $e_i$, $i =1, \cdots, n+1$ and is defined
by  the metric
\begin{equation}
\langle e_i, e_j \rangle = \ve_i \d_{ij}
\end{equation}
and the $n$-bracket relations
\begin{equation}
[e_1, \cdots, \hat{e_i}, \cdots, e_{n+1}] = (-1)^i \ve_i e_i.
\end{equation}
The signs $\ve_i$ are
given by $(+ \cdots +)$ for $\cA_{0,n+1} := \cA_{n+1}$,
$(-+ \cdots ++)$ for $\cA_{1,n}$,   $(--+\cdots +)$ for $\cA_{2,n-1}$ etc.
}
and
show that  Cartan-Weyl
3-algebras do not contain $\cA_4$ as a subalgebra in general.
This implies that a BLG
theory that is based on a Cartan-Weyl 3-algebra cannot contain fuzzy
$S^3$ in its description, at least not semiclassically.  Section 5
contains some further discussions. The detailed analysis of the consistency
conditions is performed in the appendix A.

\section{Cartan-Weyl 3-algebras and Consistency Conditions}
\label{ccw}

\subsection{Cartan-Weyl basis for a semisimple Lie algebra}

We start with some basic definitions about Lie algebras.
Let $\mathfrak{g}$ be a Lie algebra over the field $\mathbb{C}$.
A subspace $\mathfrak{a}$ is a {\it subalgebra} if
$[\mathfrak{a},\mathfrak{a}] \subset \mathfrak{a}$.  A subspace $I$ is an {\it ideal} if $[I,\mathfrak{g}]
\subset I$.
Suppose that $I_1, I_2$ are ideals of a Lie algebra $\mathfrak{g}$,
then $[I_1,I_2]$ is also an ideal of $\mathfrak{g}$.
In particular, $\mathfrak{g}$ has the following two series of ideals:
\begin{equation}
\mathfrak{g} \supset \mathfrak{g}^1 \supset \mathfrak{g}^2 \cdots,
\end{equation}
which is called the {\it descending central series}of $\mathfrak{g}$; and
\begin{equation}
\mathfrak{g} \supset \mathfrak{g}^{(1)} \supset \mathfrak{g}^{(2)} \cdots,
\end{equation}
which is called the  {\it derived series} of $\mathfrak{g}$.
Here
\begin{equation}
\mathfrak{g}^{0} := \mathfrak{g}, \quad \mathfrak{g}^{n} : =[\mathfrak{g}^{n-1},\mathfrak{g}]
\end{equation}
and
\begin{equation}
\mathfrak{g}^{(0)} := \mathfrak{g}, \quad \mathfrak{g}^{(n)} : =[\mathfrak{g}^{(n-1)},\mathfrak{g}^{(n-1)}]
\end{equation}
for $n \in \NN$.
A Lie algebra
$\mathfrak{g}$ is {\it nilpotent} if $\mathfrak{g}^{n} =0$ for some $n$. It
is {\it solvable} if $\mathfrak{g}^{(n)} =0$ for some $n$.
Obviously every nilpotent Lie algebra is also solvable.

A Lie algebra is {\it simple} if it has no proper ideals other than
itself or 0.
The union of all solvable ideal of $\mathfrak{g}$ is also a solvable ideal of
$\mathfrak{g}$, called the radical $\mathfrak{R}(\mathfrak{g})$  of $\mathfrak{g}$.  A Lie algebra is {\it
semisimple} if its radical is 0.
\if
The significance of semisimplicity is due to the {\it Levi
decomposition} which states that
\begin{thm}{(Levi decomposition)} Every finite dimensional Lie algebra $\mathfrak{g}$
admits a Levi subalgebra $\ml$, i.e. there exists
a semisimple subalgebra $\ml$ of $\mathfrak{g}$ such
that $\mathfrak{g}$ is a semidirect sum of its radical $\mathfrak{R}(\mathfrak{g})$ and $\ml$
\begin{equation}
\mathfrak{g} = \mathfrak{R}(\mathfrak{g}) \oplus \ml
\end{equation}
\end{thm}
\fi
As a result, a semisimple Lie algebra is given by a direct sum of
simple Lie algebras and Abelian ones.

In the theory of Lie algebra, a very useful device  is the {\it Killing metric}
which can be defined using the
adjoint representation ${\rm ad} : \mathfrak{g} \to \mathfrak{g}$ by
\begin{equation}
\k (x,y) = \tr \big( ({\rm ad}\, x)({\rm ad}\, y) \big), \quad \mbox{for
  $x,y \in \mathfrak{g}$ }.
\end{equation}
The Killing metric is invariant:
\begin{equation}
\k ([x,z],y) +\k(x,[y,z]) =0, \quad \mbox{for all $x,y,z \in \mathfrak{g}$}.
\end{equation}
Killing metric plays an important role in characterizing Lie algebra. In
fact,
\begin{thm}{(Cartan criteria of semisimplicity)} \label{C1}
A Lie algebra $\mathfrak{g}$ is semisimple iff the Killing metric $\k$ is non-degenerate.
\end{thm}

\begin{thm}{(Cartan criteria of solvability)} \label{C2}
A Lie algebra $\mathfrak{g}$ is solvable iff the Killing metric  $\k([x,y],z)
=0$ for all $x,y,z \in \mathfrak{g}$.
\end{thm}

It is instructive  to recall
the reason why semisimple Lie algebras play a fundamental role  in
physics.  In
fact the classification theorem of Lie groups
states that any compact, connected
Lie group is a product of a finite abelian group and
simple Lie groups \footnote{We also recall that a simple Lie group
is isomorphic to exactly one of the $SU(n), n\geq 3;  Sp(n) ,n\geq
1;  Spin(n), n\geq 7;  G_2, F_4, E_6, E_7, E_8$.}.
By modelling the continuous symmetries that appears in nature
by compact connected Lie groups, one thus arrive at the semisimple Lie
algebras naturally.

An important result in the theory of Lie algebra
is that a semisimple Lie algebra admits a
very nice basis of generators called the Cartan-Weyl basis.
A Cartan-Weyl basis consists of a set of generators $H_I$ from the
Cartan subalgebra and
a set of step generators $E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}$  that are labelled by a vector
$\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}=(\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}_I)$ called the root.
In general any two Cartan subalgebras are conjugate  relative to the group
of special automorphisms generated by the exponents of nilpotent inner
derivations \cite{jacobson,OV}.
A special feature of the Cartan-Weyl
basis is that the Cartan subalgebra is Abelian and
the roots are  non-degenerate.
In this basis, the Killing metric reads
\begin{equation} \label{metric-cA}
\langle E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}, E^\b \rangle = \d_{\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}+\b}, \quad \langle E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}, H^I \rangle =0,
\quad g_{IJ} := \langle H_I,H_J \rangle,
\end{equation}
where  $g_{IJ}$, the restriction of the Killing metric on the Cartan
subalgebra, is non-degenerate for a semisimple Lie algebra.
The Lie brackets take the form
\begin{align} \label{cw2-1}
[H_I,H_J] &=& 0, \nonumber\\
\;  [ H_I, E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A} ] &=& \alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}_I E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}, \nonumber\\
\; [E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}, E^\b] &=& \begin{cases}
0, & \mbox{if $\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}+\b \neq 0$ not  a root},\\
c(\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A},\b) E^{\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}+ \b}, & \mbox{if $\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}+\b \neq 0$ is a root},\\
- \alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A} \cdot H & \mbox{if $\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}+\b =0$},
\end{cases}
\end{align}
where $\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A} \cdot H = \alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}_I g^{IJ} H_J$ and  $g^{IJ}$ is the inverse of
$g_{IJ}$.
Here we have used the invariance of the metric in
deriving the relation for $[E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A},E^{-\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}}]$. The Cartan-Weyl basis is
specified by the system of roots and the coefficient
$c(\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A},\b)$. They can be  solved and gives a complete classification of
semisimple Lie algebras.

The proof of the existence of a Cartan-Weyl basis for a semisimple Lie
algebra rests on the theory of root space decomposition  for Lie
algebras, see for example, \cite{jacobson,OV}. We have also included a
proof in the companion paper \cite{cat2}, highlighting the most important ideas
involved so as to explain the conditions that are needed to establish the
existence of a {\it generalized Cartan-Weyl basis}
for a Lie 3-algebra.
Here let us take an elementary and more direct approach
to explain
the assumptions involved that lead to
the existence of a generalized Cartan-Weyl basis. This
exercise will also be useful for
a heuristic understanding the conditions that lead to
the existence of a similar Cartan-Weyl basis for a Lie 3-algebra.

Given a Lie algebra, let us start by looking
for a maximal set of commuting generators $H_I$, $I =1, \cdots,
N$. Denote the rest of the generators by $E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}$ where $\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}$ is just a
label at this point.
Denote also the set of all generators by $Y^a = \{ H_I,
E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}\}$. Most generally we have
\begin{equation} \label{eigen}
[H_I, Y^a] = \psi_I{}^a{}_b Y^b,
\end{equation}
where $\psi_I= (\psi_I){}^a{}_b$ is a constant matrix. Jacobi identity
$[H_I,[H_J, Y^a]] + \cdots =0 $ implies that
\begin{equation}
[\psi_I, \psi_J] = 0
\end{equation}
and so we can diagonalize $\psi_I$'s simultaneously. If one assumes that
all the
eigenvalues are non-degenerate, one obtains
$\psi_I{}^a{}_b = \d^a{}_b \psi_{Ia}$. Obviously $\psi_{Ia} =0$ for
$a=K$. Therefore
\begin{equation}
[H_I, E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}] = \alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}_I E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A},
\end{equation}
where we have denoted $\psi_{I \alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}} = \alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}_I$.
Next using the invariance of the metric $\d_{H_I}\langle E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}, E^\b \rangle =0$
and $\d_{H_I}\langle H_I, E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A} \rangle=0$, one obtains immediately
\begin{equation}
\langle E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}, E^\b \rangle \propto \d^{\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}+\b}, \quad \langle H_I, E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A} \rangle =0.
\end{equation}
Normalizing the generators suitably and assuming that the metric is
non-degenerate when restricted to the $H$'s,
we obtain the metric \eq{metric-cA}. Next we note
that the Jacobi identity gives $ [H_I,[E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A},E^\b]] = (\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}+\b)_I
[E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A},E^\b]$. Therefore, using also the invariance of the metric, one sees
that
$[E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}, E^\b] $ takes the form as in \eq{cw2-1}. This is precisely
the Cartan-Weyl basis.

What we
see from this analysis is that the existence of a Cartan-Weyl basis is
equivalent to the assumption that the solutions of the
``eigenvalue equation'' \eq{eigen}  are non-degenerate; and that the
restriction of the metric to the $H$'s
is non-degenerate. This is also
equivalent to the assumption of semisimplicity.
For Lie 3-algebras, we will now show that a Cartan-Weyl basis exists with
similar assumptions of non-degeneracy.

\subsection{Cartan-Weyl 3-algebras and the Cartan-Weyl basis}

A Lie $n$-algebra $\cA$ \cite{filippov,K1} is a linear space over a
field $\mathbb{F}$ on which defined is
a multilinear $n$-bracket operation which is skew-symmetric and
satisfies the fundamental identity:
\begin{equation}
[ [b_1, \cdots, b_n], a_1, \cdots, a_{n-1}] = \sum_{i=1}^n
[ b_1, \cdots, [b_i, a_1, \cdots, a_{n-1}], \cdots, b_n]
\end{equation}
for all $a_i, b_j \in \cA$.
For $n=2$, we get back to the usual
Lie algebra and the fundamental identity coincides
with the   Jacobi identity. We will be concerned with
real Lie $n$-algebras in this paper.

The fundamental identity guarantees that the transformation
$\d: \cA \rightarrow \cA$ defined by
\begin{equation}
\d_{(a_1,\cdots,a_{n-1})} f := [f, a_1,\cdots, a_{n-1}]
\end{equation}
is a derivation of the Lie $n$-algebra:
\begin{equation}
\d [b_1, \cdots, b_n] = [\d b_1, \cdots,b_n]+ \cdots + [b_1,\cdots,\d b_n].
\end{equation}
The map $\d$ is parametrized by a skew-symmetric
collection of $n-1$ elements  $(a)=(a_1, \cdots, a_{n-1})$ and is a
natrual generalization of
the usual gauge transformation $\d_a f:= [f,a]$ defined for Lie algebra.

A {\it metric} $\langle\cdot , \cdot \rangle : \cA \otimes \cA \rightarrow \mathbb{R}$
is a symmetric bilinear form on $\cA$ which is also invariant in the
sense that
\begin{equation}
\d_{(a)}\langle f,g \rangle = 0,
\end{equation}
i.e.
\begin{equation}
\langle [f,(a)], g\rangle + \langle f, [g,(a)]\rangle =0,
\end{equation}
for any $f,g \in \cA$ and $(a) \in \cA^{\wedge (n-1)}$.
In addition to a metric, it is also natural to introduce the notion of
an
{\it invariant form}. An invariant form on  a Lie $n$-algebra $\cA$ is a
multilinear function $F$ on $\cA^{\wedge (n-1)} \times \cA^{\wedge (n-1)}$
which satisfies the invariance condition:
\begin{align} \label{inv-form}
\sum_{i=1}^{n-1}
&& F(a_1, \cdots , a_i R_{(c)}, \cdots, a_{n-1}; b_1, \cdots, b_{n-1}) \nonumber\\
&& \qquad
+ F( a_1, \cdots, a_{n-1};b_1, \cdots, b_i R_{(c)}, \cdots, b_{n-1}) =0,
\end{align}
where $R_{(c)}$ is the right multiplication
$a R_{(c)} :=[a,c_1, \cdots, c_{n-1}]$.
We say an invariant form is non-degenerate if $F(a_1, \cdots, a_n ;b_1,
\cdots, b_{n-1}, x ) =0$ for all $a_1, \cdots, a_n, b_1, \cdots, b_{n-1}
\in \cA$ implies that $x=0$.
In this paper we will be mainly interested in Lie 3-algebra with an invariant
non-degenerate metric.  We will refer to this as a metric Lie 3-algebra.

In analogy to the Lie algebra case,
one may introduce the notion of a
Cartan-Weyl basis for a Lie 3-algebra as follows. Let $\cA$ be a metric Lie
3-algebra, we call $\cA$ a {\it Cartan-Weyl 3-algebra} if the algebra admits
a basis of generators $H_I$, $I =1, \cdots, N \geq 2$ and $E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}$, $\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}
= (\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}_{IJ})$, with the metric
\begin{equation}  \label{metric3-eeh}
\langle E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}, E^\b \rangle = \d_{\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}+\b},\quad \langle E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}, H_I \rangle =0,
 \quad g_{IJ}:= \langle  H_I, H_J \rangle \;\;\mbox{non-degenerate},
\end{equation}
and the 3-brackets:
\begin{align}
\; [H_I, H_J, H_K] &=& 0, \label{cw31} \\
\; [H_I, H_J, E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}]& =& \alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}_{IJ} E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}, \label{cw32}
\end{align}
\begin{subnumcases}{[H_I,E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}, E^\b] =}
\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}_{IK} g^{KL} H_L, & \mbox{if $\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}+ \b =0$,} \label{cw33a}\\
g_I(\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A},\b) E^{\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}+\b}, & \mbox{if $\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}+ \b \neq 0$ is a root,} \label{cw33b}\\
0,  & \mbox{if $\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}+ \b$ is not a root,} \label{cw33c}
\end{subnumcases}
\begin{subnumcases}{[E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A},E^\b, E^\g] =}
- g_K(\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A},\b) g^{KL} H_L,  & \mbox{if $\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}+ \b + \g =0$,} \label{cw34a} \\
c(\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A},\b,\g) E^{\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}+\b+\g},  & \mbox{if $\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}+ \b + \g \neq 0$ a root,}
 \label{cw34b}\\
0,  & \mbox{if $\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}+ \b + \g$ is not a root.}   \label{cw34c}
\end{subnumcases}
In writing these 3-bracket relations, we have used the invariance of the
metric to relate \eq{cw33a} with \eq{cw32} and \eq{cw34a} with \eq{cw33b}.
The number of independent $H$'s will be called the {\it rank} of the
Cartan-Weyl 3-algebra and the set of generators $\{H_I, E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}\}$
will be called a {\it Cartan-Weyl basis}. Similarly one can define a
Cartan-Weyl Lie $n$-algebra.

It is natural to ask if semisimplicity is again the right condition to
guarantee the existence of a Cartan-Weyl basis for a Lie 3-algebra
(or Lie $n$-algebra in general). It turns out that
there is a number of ways one can generalize the concept of
semisimplicity of Lie algebra to the higher
Lie $n$-algebra, and so the answer is not obvious.
In the paper \cite{cat2}, we will investigate this question and
find out what are the precise mathematical
conditions for a Lie 3-algebra to be a (generalized) Cartan-Weyl
3-algebra.

Without entering into these more abstract discussions, it is possible
to state and explain quite explicitly what are the conditions
that lead to the existence of a  Cartan-Weyl basis. In analogy to
the discussion for  Lie algebras,  let us start by saying we have a
maximal set of ``commuting'' generators
in the sense that
\begin{equation}
[H_I,H_J,H_K] =0, \quad I,J,K = 1,\cdots, N.
\end{equation}
Denote the rest of the generators by $E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}$ and the set of all
generators by $Y^a = \{ H_I, E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}\}$. Most generally we have
\begin{equation} \label{eigen2}
[H_I, H_J, Y^a] = \psi_{IJ}{}^a{}_b Y^b,
\end{equation}
where $\psi_{IJ}= (\psi_{IJ}){}^a{}_b$ is a constant
matrix. Fundamental identity $[H_K,H_L,[H_I,H_J,Y^a]] = \cdots $
implies that
\begin{equation}
[\psi_{IJ}, \psi_{KL}] = 0
\end{equation}
and so we can diagonalize $\psi_{IJ}$'s simultaneously.
If one assume that all the eigenvalues are non-degenerate, then we have
$\psi_{IJ}{}^a{}_b = \d^a{}_b \psi_{IJ\,a}$. Obviously $\psi_{IJ\,a} =0$ for
$a=K$. Denoting $\psi_{IJ\, \alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}} = \alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}_{IJ}$, we have
\begin{equation} \label{eigen1}
[H_I, H_J, E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}] = \alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}_{IJ} E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}
\end{equation}
and so the $E$-type generators are parametrized by a two-form
$\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}_{IJ}$.
Next using the invariance of the metric $\d_{(H_I,E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A})}\langle E^\b, H_J \rangle =0$
and $\d_{(H_I,H_J)}\langle E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}, H_K \rangle=0$, we obtain immediately
\begin{equation}
\langle E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}, E^\b \rangle \propto \d^{\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}+\b}, \quad \langle H_I, E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A} \rangle =0.
\end{equation}
Normalizing suitably the generators
and assuming that the metric is non-degenerate
when restricted to the $H_I$'s, we get precisely the metric
\eq{metric3-eeh}.
Next use the fundamental identity $[H_J,H_K,[H_I, E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A},E^\b]] = \cdots$,
we obtain
$[H_J,H_K,[H_I, E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A},E^\b]] = (\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}+\b)_{JK} [H_I, E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A},E^\b]
$. Together with the assumption that the $H_I$'s  forms a maximal
set of commuting generators, we obtain
$[H_I, E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A},E^\b] =c_I(\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}){}^J H_J$  for $\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}+ \b=0$. Invariance of the
metric then implies that it is precisely of the form \eq{cw33a}.
For $\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}+\b \neq 0$, if we denote
\begin{equation}
[H_I, E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A},E^\b] = \l_I{}^{\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}\b}{}_a Y^a,
\end{equation}
then the fundamental identity $[[H_I,H_J,H_K],E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A},E^\b] = \cdots$ gives
\begin{equation} \label{lc1}
\sum_{ \mbox{$(I,J,K)$ cyclic}}
 \l_I{}^{\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}\b}{}_\g \g_{JK} =0.
\end{equation}
On the other hand, the invariance of the metric
$\d_{(H_I,E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A})} \langle E^\b,E^\g \rangle =0$ implies
that
\begin{equation} \label{lc2}
 \l_I{}^{\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}\b\g} = \l_I{}^{[\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}\b\g]}
\end{equation}
is completely antisymmetric in $\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A},\b, \g$. Here we have used the metric
$\langle E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A},E^\b \rangle = \d^{\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}+\b} $ to raise an indices of $\l$.
Using \eq{lc1} and \eq{lc2} and the fundamental identity
$[[H_i,H_j,E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}],H_K,E^\b] = \cdots$, one can easily deduce that
$\l_I{}^{\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}\b\g} = 0$ unless $\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}+\b+\g =0$. Therefore
$\l_I{}^{\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}\b\g}$ takes the form
\begin{equation}
\l_I{}^{\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}\b\g} = g_I(\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A},\b), \quad \mbox{for $\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}+\b+\g =0$}
\end{equation}
and so we obtain the 3-bracket \eq{cw33b}.
Finally, assume that
\begin{equation}
[E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A},E^\b, E^\g] = \o^{\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}\b\g}{}_a Y^a.
\end{equation}
By considering the
fundamental identity $[[E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A},E^\b,E^\g],H_J,H_K] = \cdots$, one can show that
\begin{equation} \label{omega}
(\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}+\b+\g+\d) \o^{\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}\b\g\d} =0, \quad
(\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}+\b+\g)  \o^{\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}\b\g L} =0.
\end{equation}
The first condition implies that $\o^{\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}\b\g\d} =0$ unless $\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}+\b+\g+\d =0$.
In this case, by denoting $\o^{\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}\b\g\d} = c(\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A},\b,\g)$, we arrive at
the 3-bracket \eq{cw34b} after using the second condition of
\eq{omega}. The second condition of \eq{omega}
gives precisely the 3-brackets \eq{cw34a} after using
the invariance of the metric.

All in all, this analysis demonstrates that
the existence of a Cartan-Weyl basis for a Lie 3-algebra is
equivalent to the requirement that
``eigenvalue equation'' \eq{eigen2}  has non-degenerate solutions;
and that the
restriction of the metric to the $H$'s
is non-degenerate.

\subsection{Consequences of the consistency conditions}

A Cartan-Weyl 3-algebra is specified by the data:
the roots $\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}_{IJ}$ and the structural constants
$g_I(\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A},\b)$ and $c(\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A},\b,\g)$.
Remarkably, the consistency conditions arising from the fundamental
identities are so strong that one can solve them exactly, giving a
complete classification of
Cartan-Weyl 3-algebras.

The consistency conditions that follow from the fundamental
identities
are listed and analyzed in details in the
appendix. It turns out  that we have generally
\begin{equation}
\boxed{
c(\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A},\b,\g) =0.
}
\end{equation}
As for the roots, they generally have to satisfy the condition \eq{cond1}.
For a Cartan-Weyl
3-algebra with a
single pair of roots $\pm \alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}$,  \eq{cond1} is the only condition to be
satisfied.
For Cartan-Weyl 3-algebra with more than one pair of roots,
the root space can generally be decomposed into a number
of (say $M$) components: $\D_\cA(\cH) = \oplus_{{\mbox{{\suetterlin i}}}=1}^M \Omega_{\mbox{{\suetterlin i}}}$,
where there is a null
vector $\ph^{({\mbox{{\suetterlin i}}})}$ associated with each $\Omega_{\mbox{{\suetterlin i}}}$. The roots in each
$\Omega_{\mbox{{\suetterlin i}}}$ can be decomposed in the form
\begin{equation} \label{cond-ap}
\boxed{
\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A} = \ph^{({\mbox{{\suetterlin i}}})} \wedge {\hat{\a}}^{({\mbox{{\suetterlin i}}})}, \quad \alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A} \in \Omega_{\mbox{{\suetterlin i}}}.
}
\end{equation}
The corresponding
one-form parts ${\hat{\a}}^{({\mbox{{\suetterlin i}}})}$ form the root system of a semisimple Lie
algebra $\mathfrak{g}^{({\mbox{{\suetterlin i}}})}$.
Moreover $\ph^{({\mbox{{\suetterlin i}}})}$ and ${\hat{\a}}^{({\mbox{{\suetterlin j}}})}$ satisfy the conditions
\begin{equation} \label{cond6ap}
\boxed{
\ph^{({\mbox{{\suetterlin i}}})} \cdot \ph^{({\mbox{{\suetterlin j}}})} =0,
\quad \mbox{for all ${\mbox{{\suetterlin i}}},{\mbox{{\suetterlin j}}}$},
}
\end{equation}
\begin{equation} \label{cond6p}
\boxed{\ph^{({\mbox{{\suetterlin i}}})} \cdot {\hat{\a}}^{({\mbox{{\suetterlin j}}})} =0,
\quad \mbox{for all ${\mbox{{\suetterlin i}}},{\mbox{{\suetterlin j}}}$},
}
\end{equation}
\begin{equation}\label{cond7p} \boxed{
{\hat{\a}}^{({\mbox{{\suetterlin i}}})} \cdot {\hat{\b}}^{({\mbox{{\suetterlin j}}})} =0, \quad \mbox{for ${\mbox{{\suetterlin i}}} \neq {\mbox{{\suetterlin j}}}$},
}
\end{equation}
where the dot product is taken with respect to the metric $g_{IJ}$ of the
Lie 3-algebra.
As for $g_I(\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A},\b)$, we find
\begin{equation} \label{cond-gI}
\boxed{
g_I (\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A},\b)  = \begin{cases}
\ph_I^{({\mbox{{\suetterlin i}}})} c^{({\mbox{{\suetterlin i}}})}({\hat{\a}}^{({\mbox{{\suetterlin i}}})},{\hat{\b}}^{({\mbox{{\suetterlin i}}})}), & \quad \mbox{for $\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A},\b \in
\Omega_{\mbox{{\suetterlin i}}}$}, \\
0, & \quad \mbox{otherwise}.
\end{cases}
}
\end{equation}
where the coefficients $c^{({\mbox{{\suetterlin i}}})}$
specifies the ``$[E,E]=E$''  type brackets of
the semisimple Lie algebra $\mathfrak{g}^{({\mbox{{\suetterlin i}}})}$
as in \eq{lie-cw}.
\if
Finally the coefficient $c(\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A},\b)$ is given by
\begin{equation} \label{cond-cp}
\boxed{
c(\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A},\b) = \begin{cases}
c^{({\mbox{{\suetterlin i}}})}({\hat{\a}}^{({\mbox{{\suetterlin i}}})},{\hat{\b}}^{({\mbox{{\suetterlin i}}})}),
& \mbox{$\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A},\b \in \Omega_{\mbox{{\suetterlin i}}}$, }
 \\
0, & \mbox{otherwise}.
\end{cases}
}
\end{equation}
\fi

The construction and classification of Cartan-Weyl
3-algebras has thus been reduced  to the problem of constructing
the null vectors $\ph^{({\mbox{{\suetterlin i}}})}$ and roots ${\hat{\a}}^{({\mbox{{\suetterlin i}}})}$ that satisfy the
conditions  \eq{cond6ap}-\eq{cond7p}. This can be done fully
and explicitly. This will be our task in the next section.

\section{Explicit Classification of Cartan-Weyl 3-algebras}\label{explicit}

Let us first consider the case of a
rank $N$ Cartan-Weyl 3-algebra with only a
single pair of roots $\pm \alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}$.
In this case, the condition \eq{cond1} is the only solution to be
satisfied. The general solution of it is
\begin{equation}
\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A} = \alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}_1 \wedge \alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}_2,
\end{equation}
where $\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}_1, \alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}_2$ are linearly independent. Given $\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}_1, \alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}_2$, one can
find a basis of $N$ vectors $p_1, p_2, q_3, \cdots q_N$
such that  $q_l \cdot \alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}_1 = q_l \cdot \alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}_2 =0$ and $p_1$, $p_2$ lies
in the plane spanned by $\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}_1$ and $\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}_2$. It follows immediately
that
\begin{align}
\; && [ q_l\cdot H, H_I, E^{\pm\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}}] = 0, \nonumber\\
\; && [ q_l\cdot H, E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}, E^{-\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}}] =0, \quad l =3, \cdots, N.
\end{align}
Thus the $N-2$ generators $q_l\cdot H$ are central in the algebra and
the nontrivial part of the Lie 3-algebra
has  only 4 generators $\{ p_1 \cdot H, p_2 \cdot
H, E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}, E^{-\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}} \}$. Depending on the signature of the vector space
spanned by the vectors $p_1, p_2$, this 3-algebra is isomorphic to
$\cA_{0,4}, \cA_{1,3}$ or $\cA_{2,2}$.

Therefore  let us consider the general case where the
Cartan-Weyl 3-algebra
has more than a single pair of root.
Since our results given in the section 2.2 relies on the
existence of the null vectors $\ph^{({\mbox{{\suetterlin i}}})}$, therefore it is natural to
classify the
Cartan-Weyl 3-algebra according to the number of negative
eigenvalues of the metric $g_{IJ} := \langle H_I, H_J \rangle$. With a Hermitian
structure
\begin{equation}
(H_I)^\dag =H_I, \quad (E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A})^\dag = E^{-\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}},
\end{equation}
this is also the same as the index of
the metric over the whole Lie 3-algebra $\cA$.

\subsection{Index  0 and 1}

For index 0,   Cartan-Weyl 3-algebra
is possible only if the 3-algebra has
a single pair of root. In this case the 3-algebra is isomorphic to
$\cA_{0,4}$ plus a number of central elements. This is decomposable.
Obviously it is sufficient to classify indecomposable   Cartan-Weyl
3-algebra. In the following we will
consider only indecomposable   Cartan-Weyl 3-algebra.

Next consider the case of index 1. Let us choose a basis of the Cartan
subalgebra as $\{H_I\} = \{ H_\Ih,H_a\}$, $\Ih=1, \cdots, \mathfrak{N}$, $ a=1,2$,
such that   the metric
takes the form
\begin{equation}
g_{I J} =
\begin{pmat} ( {|.} )
g_{\Ih \Jh} &   &  \cr \-
 & 1 & \cr
 & & -1 \cr
\end{pmat},
\end{equation}
where  $g_{\Ih \Jh}$ is Euclidean. Here we have the rank  $N= \mathfrak{N}+2$.

\noindent \underline{root system}

To construct a    Cartan-Weyl 3-algebra, we
need to pick ${\hat{\a}}^{({\mbox{{\suetterlin i}}})}$ and null vectors $\ph^{({\mbox{{\suetterlin i}}})}$
such that \eq{cond6ap}-\eq{cond7p} are satisfied. Knowing already that
${\hat{\a}}^{({\mbox{{\suetterlin i}}})}$ are roots of a Lie algebra, the most general
solution is to take $({\hat{\a}}_I^{({\mbox{{\suetterlin i}}})}) = ({\hat{\a}}_\Ih^{({\mbox{{\suetterlin i}}})}, 0, 0)$ and to take
the null vectors in the orthogonal complements $\ph^{({\mbox{{\suetterlin i}}})} = ({\bf 0},
\cdot, \cdot)$. There can be (up to overall normalization)
two choices of such null vectors. But since they cannot be orthogonal
to each other, it means we can only use one  null vector in the index
1 case. Without loss of generality, let us take
\begin{equation}
\ph= ({\bf 0}, 1, 1)/\sqrt{2},
\end{equation}
and the roots of the   Cartan-Weyl 3-algebra are of the form
\begin{equation}
\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A} = \ph \wedge {\hat{\a}}, \quad \mbox{where}\quad ({\hat{\a}}_I)= ({\hat{\a}}_\Ih,0,0) .
\end{equation}
Explicitly, it has the nonvanishing components $\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}_{a \Ih}= \ph_a {\hat{\a}}_\Ih$.

\noindent\underline{3-brackets}

It is convenient to reorganize the basis of Cartan generators as
\begin{equation}
H_I = \{ H_\Ih, v, u \},
\end{equation}
where
\begin{equation}
v:= \ph \cdot H, \quad u:= \pt \cdot H
\end{equation}
with
\begin{equation}
\pt := ({\bf 0}, 1, -1)/\sqrt{2}.
\end{equation}
It is
\begin{equation} \label{i1-0}
\langle v,v\rangle = \langle u, u \rangle =0 \quad \mbox{and $\langle v,u\rangle =\ph \cdot \pt
=1$}.
\end{equation}
Since the ``step''
generators $E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}$ are characterized by the one-form part ${\hat{\a}}$,
it can be written as $E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A} = E^{\hat{\a}}$.
As a vector space,
the Cartan-Weyl 3-algebra of index 1 is given by
\begin{equation}
\cA = \mathfrak{g} \oplus \mathbb{C}(u,v).
\end{equation}
The   Cartan-Weyl 3-algebra relations read
\begin{align}
\; &&[v, \cdot, \cdot]= 0, \label{i1-1}\\
\;&& [u,H_\Ih, E^{\hat{\a}}] = {\hat{\a}}_\Ih E^{\hat{\a}}, \label{i1-2}\\
\; && [u, E^{\hat{\a}}, E^{\hat{\b}}] = \begin{cases}
{\hat{\a}}_\Ih g^{\Ih\Jh} H_\Jh, & {\hat{\a}}+{\hat{\b}}=0, \\
c({\hat{\a}},{\hat{\b}})E^{{\hat{\a}}+{\hat{\b}}}, &  \mbox{${\hat{\a}}+{\hat{\b}}$ nonzero roots},
\end{cases} \label{i1-3}\\
\; && [E^{\hat{\a}}, E^{\hat{\b}}, E^{\hat{\g}}] = \begin{cases}
-c({\hat{\a}},{\hat{\b}}) v, & {\hat{\a}}+{\hat{\b}}+{\hat{\g}} =0, \\
0, & \mbox{otherwise}.
\end{cases} \label{i1-4}
\end{align}
The relations \eq{i1-1} - \eq{i1-4} can be summarized as
\begin{align} \label{i1-5}
\; && [v, \cdot, \cdot]= 0, \nonumber \\
\; && [u, g_1,g_2]  = [g_1,g_2]_\mathfrak{g}, \\
\; && [g_1,g_2,g_3] = -\langle [g_1,g_2], g_3 \rangle_\mathfrak{g} v , \nonumber
\end{align}
where $g_i \in \mathbb{C}(H_I, E^{\hat{\a}}):=\mathfrak{g}$ is a semisimple Lie algebra with
the  brackets $[\cdot,\cdot]_\mathfrak{g}$
and the metric   $\langle \cdot , \cdot \rangle_\mathfrak{g}$
as defined by \eq{metric-cA}.

We note that the Cartan-Weyl 3-algebra of index 1
is precisely the same as  the
Lorentzian 3-algebra \cite{LOR}.

\subsection{Index 2}

For index 2,
let us choose a basis of the Cartan generators as $\{H_I\}= \{H_\Ih, H_a
\}$, $\Ih = 1, \cdots, \mathfrak{N}$, $a=1, \cdots, 4$,
such that the metric takes the form with nonzero entries
\begin{equation} \label{m2}
\begin{pmat} ( {|.} )
g_{\Ih \Jh} &   &  & &	\cr\-
 & 1 &   &  &         	\cr
 &   & -1&  & 		\cr
 &   &   & 1& 		\cr
 &   &   &  & -1        \cr
\end{pmat},
\end{equation}
where  $g_{\Ih \Jh}$ is Euclidean. In this case, the rank  $N= \mathfrak{N}+4$.

\noindent\underline{root system}

We can solve the conditions \eq{cond6p} by
taking $\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}_I =(\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}_\Ih; 0,0,0,0)$ and the null vectors in the
complementary part of the vector
space.   There are four such linearly independent  null vectors.
One can choose the basis of null
vectors $\{ \ph^{(1)}, \ph^{(2)}, \pt^{(1)},\pt^{(2)} \}$ such that
$\ph^{({\rm i})} \cdot \ph^{({\rm j})} = 0$,
$\pt^{({\rm i})} \cdot \pt^{({\rm j})} = 0$,
$\ph^{({\rm i})} \cdot \pt^{({\rm j})} = \d_{{\rm i}{\rm j}}$
\footnote{
For example, with respect to the
metric \eq{m2}, one can take
$\ph^{(1)}= ({\bf 0}, 1,1,0,0)/\sqrt{2}$,
$\ph^{(2)}= ({\bf 0}, 0,0, 1,1)/\sqrt{2}$,
$\pt^{(1)}= ({\bf 0}, 1,-1,0,0)/\sqrt{2}$,
$\pt^{(2)}= ({\bf 0}, 0,0, 1,-1)/\sqrt{2}$.
However we will not need
these explicit expressions.
}.
Moreover since any linear combinations of the
$\ph$-vectors (or of the $\pt$-vectors) are still null,
the most general set (up to trivial overall normalization factor)
of null vectors which satisfy \eq{cond6ap} is thus given by
\begin{equation} \label{p1s}
\ph^{(\l)} := \ph^{(1)} + \l \, \ph^{(2)},
\end{equation}
where $\l$ is an arbitrary nonvanishing finite
constant. One may use as many choice of $\l$ as
one like. Denote such a set as $\L$.  The set of usable null vectors
is thus given by $\{ \ph^{({\mbox{{\suetterlin i}}})}\} = \{ \ph^{({\rm i})}, \ph^{(\l)} \}$
with ${\rm i} =1,2$ and $\l \in \Lambda$.

To proceed, let us
further partition $(\Ih)$ into
\begin{equation}
(\Ih) = (\Ih_1, \Ih_2, (\Ih_\l)_{\l\in\L} ),
\end{equation}
such that $\Ih_{\rm i} = 1, 2, \cdots, N_{\rm i}$,  $\Ih_\l = 1, 2, \cdots, N_\l$
 and $\sum_{\rm i} N_{\rm i} + \sum_\l N_\l =\mathfrak{N}$.
The set of Cartan generators $H_\Ih$ can be relabelled as
\begin{equation}
\{H_\Ih \} = \{ H_{\Ih_1}, H_{\Ih_2},  \{ H_{\Ih_\l}\}_{\l\in\L}  \}.
\end{equation}
Corresponds to each subset $\{H_{\Ih_{\rm i}} \}$ or $\{ H_{\Ih_\l}\}$,
one can associate a
set of 1-form roots such that they have nonzero components only when their
indices are in $\Ih_{\rm i}$ or $\Ih_\l$:
\begin{align} \label{1roots-a}
&& {\hat{\a}}^{({\rm i})}_I = (0, \cdots,0, {\hat{\a}}^{({\rm i})}_{\Ih_{\rm i}},0,
\cdots, 0;\, 0,0,0,0), \quad {\rm i}=1, 2,\\
&& {\hat{\a}}^{(\l)}_I = (0, \cdots,0, {\hat{\a}}^{(\l)}_{\Ih_\l},0, \cdots, 0;\,
0,0,0,0), \quad \l \in \Lambda \label{1roots-a2}
\end{align}
The last four entries refer to the ``internal'' indices $a=1,2,3,4$.
The 1-forms \eq{1roots-a}, \eq{1roots-a2}
satisfy the conditions \eq{cond6p} and \eq{cond7p} by
construction and lead to the  following set of two-form roots:
\begin{align}
&& \alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}^{({\rm i})}= \ph^{({\rm i})} \wedge {\hat{\a}}^{({\rm i})}, \quad {\rm i} =1,2,  \label{2roots-a}\\
&& \alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}^{(\l)}= \ph^{(\l)} \wedge {\hat{\a}}^{(\l)}, \quad \l \in \L. \label{2roots-b}
\end{align}
Or, in terms of the nonzero components, we have
\begin{align}
&& \alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}^{({\rm i})}_{a \Ih_{\rm i}} = \ph^{({\rm i})}_a {\hat{\a}}^{({\rm i})}_{\Ih_{\rm i}}\;, \quad{\rm i} =1,2,\\
&& \alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}^{(\l)}_{a \Ih_\l} = \ph^{(\l)}_a {\hat{\a}}^{(\l)}_{\Ih_\l},  \quad \l \in \L.
\end{align}

In addition to these roots which have a ``mixed'' indices structure,
one can also consider  roots with ``internal'' indices. In particular to
satisfy \eq{cond6p} and \eq{cond7p}, the only possibility is have a
root which is a linear combination of $\ph^{(1)}$ and $\ph^{(2)}$. Wedging
it with \eq{p1s} leads to the 2-form roots
\begin{equation} \label{2roots-r}
r^{(n)} = c_n \ph^{(1)} \wedge \ph^{(2)},
\end{equation}
where $c_n$ is a constant and $n$ is taken from an arbitrary set $X$.
 It is easy to show that
$c(r^{(n)}, r^{(m)}) =0$ for all $n,m \in X$.

All in all, we obtain the roots \eq{2roots-a}, \eq{2roots-b}
and \eq{2roots-r}. This
give rises to the step generators
\begin{equation}
E^{\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}^{({\rm i})}}, E^{\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}^{(\l)}}, E^{\pm r^{(n)}}.
\end{equation}

Next let us regroup the generators of the
Cartan-Weyl 3-algebra in a way which
will be convenient for our analysis. Let us start with the
set of Cartan generators
\begin{equation}
\{H_I\} = \{ H_{\Ih_1}, H_{\Ih_2},  \{ H_{\Ih_\l}\}_{\l\in\L} , H_a  \}.
\end{equation}
To express the 3-brackets, it is convenient to use
the following 4 generators instead of the $H_a$'s:
\begin{equation}
v^{({\rm i})} := \ph^{({\rm i})} \cdot H, \quad
u^{({\rm i})} := \pt^{({\rm i})} \cdot H,\qquad {\rm i} =1,2.
\end{equation}
It is $\langle v^{({\rm i})},  v^{({\rm j})} \rangle =0$,
$\langle u^{({\rm i})},  u^{({\rm j})} \rangle =0$,  $\langle v^{({\rm i})},  u^{({\rm j})} \rangle
=\d_{{\rm i} {\rm j}}$.
Next we denote the generators
\begin{equation}
 x_\pm^{(n)}:= E^{\pm r^{(n)}},
\end{equation}
where the metric is $\langle x_+^{(n)}, x_-^{(m)} \rangle = \d_{nm}$.
It is also convenient to group the step generators and the Cartan
generators in the following manners and introduce the vector spaces
\begin{equation}
\mathfrak{g}^{({\rm i})} := \mathbb{C}(H_{\Ih_{\rm i}}, E^{{\hat{\a}}^{({\rm i})}} ), \quad
\mathfrak{g}^{(\l )} := \mathbb{C}(H_{\Ih_\l }, E^{{\hat{\a}}^{(\l )}} ).
\end{equation}
The  Cartan-Weyl 3-algebra of index 2 is thus given by
\begin{equation} \label{vec2}
\cA= \bigg(\bigoplus_{{\rm i}=1}^2 \mathfrak{g}^{({\rm i})} \bigg) \oplus
\bigg( \bigoplus_{\l \in \L} \mathfrak{g}^{(\l)} \bigg) \oplus
\mathbb{C}(u^{(1)}, v^{(1)}) \oplus \mathbb{C}(u^{(2)}, v^{(2)})
\oplus E
\end{equation}
as vector space. Here $E$ is an even dimensional vector space spanned
by the elements $x_\pm^{(n)}$, $n \in X$.
Different components of the direct sum
in \eq{vec2} are orthogonal to each other. The presence of the vector
space $E$ is a new feature when the index is higher than 1.

\noindent\underline{3-brackets}

Now let us express the
3-algebra relations \eq{cw31}-\eq{cw34c} in terms of these generators.
Using our results \eq{cond5p} and the result that $c^{({\mbox{{\suetterlin i}}})}$ are
those coefficients for a semisimple Lie algebra, we obtain immediately
the following nonvanishing 3-brackets:
\begin{equation} \label{fr0}
[u^{({\rm i})}, g_1, g_2] = \begin{cases}
[g_1,g_2]_{\mathfrak{g}^{({\rm i})}}, & g_1, g_2 \in \mathfrak{g}^{({\rm i})},\\
\l_{\rm i} [g_1,g_2]_{\mathfrak{g}^{(\l)}}, &  g_1, g_2 \in \mathfrak{g}^{(\l)},
\end{cases}
\end{equation}
where $\l_1 =1$ and $\l_2= \l$, and
\begin{equation} \label{fr1}
[g_1,g_2,g_3] = \begin{cases}
- \langle [g_1,g_2],g_3 \rangle_{\mathfrak{g}^{({\rm i})}} \; v^{({\rm i})},
& g_1, g_2, g_3 \in \mathfrak{g}^{({\rm i})},\\
- \langle [g_1,g_2],g_3 \rangle_{\mathfrak{g}^{(\l)}} \; (v^{(1)} + \l v^{(2)}),
& g_1, g_2, g_3 \in \mathfrak{g}^{(\l)}.
\end{cases}
\end{equation}
Also we have
\begin{align}
&& [u^{(1)}, u^{(2)},  x_+^{(n)}] = c_n  x_+^{(n)}, \quad
 [u^{(1)}, u^{(2)},  x_-^{(n)}] = -c_n  x_-^{(n)},  \label{fr2}\\
&& [u^{(1)}, x_+^{(n)}, x_-^{(n)}] = c_n v^{(2)},\quad
[u^{(2)}, x_+^{(n)}, x_-^{(n)}] = -c_n v^{(1)}. \label{fr2a}
\end{align}

One may also rewrite \eq{fr2} and \eq{fr2a} in terms of a different basis.
Introducing $x_\pm^{(n)}= (x^{(n)}  \pm i y^{(n)} )/\sqrt{2}$.
The metric is
$\langle x^{(n)}, x^{(m)} \rangle = \langle y^{(n)},  y^{(m)} \rangle = \d_{nm}$,
$\langle x^{(n)}, y^{(m)} \rangle =0$.
The relations \eq{fr2} and \eq{fr2a} can be
rewritten in the form
\begin{align} \label{fr3}
&& [u^{(1)}, u^{(2)},x] = Jx, \nonumber \\
 && [u^{(1)},x,y] = \langle Jx,y\rangle v^{(2)},\\
 && [u^{(2)},x,y] = -\langle Jx,y\rangle v^{(1)}, \nonumber
\end{align}
where $x,y \in E$ in general and $J$ is an $so(E)$ matrix such that $J
x^{(n)} = i c_n y^{(n)}, J y^{(n)} = -i c_n x^{(n)}$.

Quite amazingly, the relations
\eq{fr0}, \eq{fr1} and \eq{fr3} are precisely the same as the
3-algebras obtained in \cite{Jind2} for the index 2 case. There the
3-algebra was constructed by requiring it to have a maximally isotropic center.
Here we have obtained the same Lie 3-algebra from a different requirement as
a Cartan-Weyl 3-algebra.

Finally we remark that in addition to the way \eq{m2} in splitting the
metric, we can also split it differently
\begin{equation}
g_{I J} =
\begin{pmat} ( {..|.} )
 &   &  & &	\cr
 & g_{\tilde{I} \tilde{J}} &   &  & \cr
 &   & &  & 		\cr \-
 &   &   & 1& 		\cr
 &   &   &  & -1        \cr
\end{pmat},
\end{equation}
such that  a (1,1) subspace is singled out separately and the metric
$g_{\tilde{I} \tilde{J}}$ is Lorentzian. Associated with the (1,1) subspace is a
null vector which one can use to construct the two-form roots.
The construction is exactly the same as for the
the index 1 case except that $g_{\tilde{I} \tilde{J}}$ is now non-Euclidean.
One obtain immediately the Lorentzian 3-algebra $\cA= \mathfrak{g} \oplus \mathbb{C}(
u,v)$
with $\mathfrak{g}$ being a semisimple Lie algebra of index 1. This is nothing
new.

\subsection{Higher index $m \geq 3$}

The above construction can be generalized to  index 3 or higher
easily. First let us choose a basis of the Cartan generators
such that the metric takes the form with nonzero entries
\begin{equation}
g_{I J} =
\begin{pmat} ( {|.} )
g_{\Ih \Jh} &     \cr \-
 & l_{ab}  \cr
\end{pmat},
\end{equation}
where $g_{\Ih \Jh}$ is
Euclidean and $l_{ab}$ is a 2$m$-dimensional
metric with signature $(m,m)$. The set of Cartan
generators is $\{ H_I \} = \{ H_\Ih ; H_a\}$ with $ \Ih =1, \cdots, \mathfrak{N}$;
$a, =1,
\cdots, 2m$. The rank is $N=\mathfrak{N}+2m$.

\noindent \underline{root system}

As before, to solve \eq{cond6p}, we can
take the 1-form roots $\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}_I =(\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}_\Ih; {\bf 0} )$
and the null vectors in the orthogonal
space: $\ph =({\bf 0}; \cdot)$.  There are $2m$ such linearly independent
null vectors. One can choose a basis of null
vectors $\{ \ph^{({\rm i})}, \pt^{({\rm i})}\}$, ${\rm i} =1, \cdots, m$ such that
\begin{equation}
\ph^{({\rm i})} \cdot \ph^{({\rm j})} = 0, \quad
\pt^{({\rm i})} \cdot \pt^{({\rm j})} = 0, \quad
\ph^{({\rm i})} \cdot \pt^{({\rm j})} = \d_{{\rm i}{\rm j}}, \qquad
{\rm i}, {\rm j}  =1, \cdots, m.
\end{equation}
In addition to the $\ph^{({\rm i})}$'s, one can also solve
\eq{cond6a} and \eq{cond6} with the following null vectors
\begin{equation}
\ph^{(\l)} := \sum_{\rm i} \l_{\rm i} \ph^{({\rm i})},
\end{equation}
where $\l_{\rm i}$'s are arbitrary constants.
One can use as many choice of $(\l_{\rm i})$ as one like. Denotes such a set as
$\Lambda$. The set of usable null vectors
is given by $\{ \ph^{({\mbox{{\suetterlin i}}})}\} = \{ \ph^{({\rm i})}, \ph^{(\l)} \}$
with ${\rm i} =1,\cdots, m$ and $(\l_{\rm i}) \in \Lambda$.

Using these null vectors, one can construct solution to \eq{cond6p} and
\eq{cond7p} by  first partitioning $(\Ih)$ into many parts
\begin{equation}
(\Ih) = ((\Ih_{\rm i})_{{\rm i} =1, \cdots, m} \; , (\Ih_\l)_{\l \in \Lambda} ),
\end{equation}
such that $\Ih_{\rm i} = 1, 2, \cdots, N_{\rm i}$ for ${\rm i}=1, \cdots,m$,
 $\Ih_\l= 1, 2, \cdots, N_\l$ and $\sum_{\rm i} N_{\rm i} +\sum_\l N_\l =\mathfrak{N}$.
The set of Cartan generators $H_\Ih$ can be relabelled as
\begin{equation}
\{H_\Ih \} = \{ \{H_{\Ih_{\rm i}}\}_{{\rm i} =1, \cdots, m} \;,
\{H_{\Ih_\l}\}_{\l \in \Lambda}  \}.
\end{equation}
Corresponds to each subset $\{H_{\Ih_{\rm i}} \}$, one can associate a
set of 1-form roots such that they have nonzero components only when their
indices are in $\Ih_{\rm i}$ or $\Ih_\l$
\begin{align}
&& {\hat{\a}}^{({\rm i})}_I = (0, \cdots,0, {\hat{\a}}^{({\rm i})}_{\Ih_{\rm i}},0, \cdots, 0;\,
{\bf 0}),    \label{1roots-ap1} \\
&& {\hat{\a}}^{(\l)}_I = (0, \cdots,0, {\hat{\a}}^{(\l)}_{\Ih_\l},0, \cdots, 0;\,
{\bf 0}). \label{1roots-ap2}
\end{align}
The last entries refers to the ``internal'' indices $a=1, \cdots, 2m$.
The set \eq{1roots-ap1}, \eq{1roots-ap2}
lead to the  following set of two-form roots:
\begin{align}
\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}^{({\rm i})}&=&  \ph^{({\rm i})} \wedge {\hat{\a}}^{({\rm i})}, \quad {\rm i} =1
,\cdots, m,\label{2roots-ap1} \\
\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}^{(\l)}&=& \ph^{(\l)} \wedge {\hat{\a}}^{(\l)}, \quad (\l) \in \Lambda .
\label{2roots-ap2}
\end{align}

In addition to these roots which have a ``mixed'' indices structure,
one can also consider  roots $\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}_{ab}$ with only ``internal'' indices.
In particular to
satisfy \eq{cond6p} and \eq{cond7p}, the only possibility is to use 1-form
roots which are linear combination of $\ph^{({\rm i})}$. Wedging
it with $\ph^{({\rm i})}$ or $\ph^{(\l)}$ leads to the 2-form roots
of the form
\begin{equation} \label{2roots-rp}
r^{(\mu)} = \sum_{{\rm i},{\rm j}} \mu_{{\rm i} {\rm j}}\;  \ph^{({\rm i})} \wedge \ph^{({\rm j})},
\end{equation}
where $\m_{{\rm i} {\rm j}}$ are  constants and not all equal to zero. There
is an internal root \eq{2roots-rp} corresponds to
each  choice of $(\mu_{{\rm i} {\rm j}})$. Denote the set of $(\m)$ by $X$.
It easy to see that one should take
$c(r^{(\m)}, r^{(\m')}) =0$  for all $(\m),(\m') \in X$.

The roots  \eq{2roots-ap1}, \eq{2roots-ap2} and \eq{2roots-rp}
give rises to the step generators
\begin{equation}
E^{\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}^{({\rm i})}},  E^{\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}^{(\l)}}, E^{\pm r^{(\mu)}}.
\end{equation}

As before, let us regroup the generators of the
3-algebra in a way which
will be convenient for our analysis. Let us start with the
set of Cartan generators. It is convenient to use
the following $2m$ generators instead of the $H_a$'s:
\begin{equation}
v^{({\rm i})} := \ph^{({\rm i})} \cdot H, \quad
u^{({\rm i})} := \pt^{({\rm i})} \cdot H,\qquad {\rm i} =1, \cdots, m.
\end{equation}
It is $\langle v^{({\rm i})},  v^{({\rm j})} \rangle =0$,
$\langle u^{({\rm i})},  u^{({\rm j})} \rangle =0$,  $\langle v^{({\rm i})},  u^{({\rm j})} \rangle
=\d_{{\rm i} {\rm j}}$.
Next we introduce the notations
\begin{equation}
 x_\pm^{(\m)}:= E^{\pm r^{(\m)}},
\end{equation}
where the metric is $\langle x_+^{(\m)},  x_-^{(\m')} \rangle = \d_{\m \m'}$.
It is also convenient to group the step generators and the Cartan
generators in the following manners and introduce the vector spaces
\begin{equation}
\mathfrak{g}^{({\rm i})} := \mathbb{C} (H_{\Ih_{\rm i}}, E^{{\hat{\a}}^{({\rm i})}} ), \quad
\mathfrak{g}^{(\l)} := \mathbb{C} (H_{\Ih_\l}, E^{{\hat{\a}}^{(\l)}} ).
\end{equation}
The  Cartan-Weyl 3-algebra of index $m \geq 3$
is thus given by
\begin{equation} \label{vec2p}
\cA= \bigg(\bigoplus_{{\rm i}=1}^m \mathfrak{g}^{({\rm i})} \bigg) \oplus
\bigg(\bigoplus_{\l \in \Lambda} \mathfrak{g}^{(\l)} \bigg) \oplus
\bigg( \bigoplus_{{\rm i}=1}^m \mathbb{C}(u^{({\rm i})}, v^{({\rm i})}) \bigg) \oplus
E
\end{equation}
as vector space. Here $E$ is a even dimensional vector space spanned
by the elements $x_\pm^{(\m)}, (\mu) \in X$.
Different components of the direct sum
in \eq{vec2p} are orthogonal to each other.

\noindent\underline{3-brackets}

The 3-brackets can be worked out similarly and we obtain the following
nonvanishing relations for the  Cartan-Weyl 3-algebra:
\begin{equation} \label{gen31}
[u^{({\rm i})}, g_1, g_2] = \begin{cases}
[g_1,g_2]_{\mathfrak{g}^{({\rm i})}}, & g_1, g_2 \in \mathfrak{g}^{({\rm i})},\\
\l_{\rm i} [g_1,g_2]_{\mathfrak{g}^{(\l)}}, &  g_1, g_2 \in \mathfrak{g}^{(\l)},
\end{cases}
\end{equation}
\begin{equation}\label{gen32}
[g_1,g_2,g_3] = \begin{cases}
- \langle [g_1,g_2],g_3 \rangle_{\mathfrak{g}^{({\rm i})}} \; v^{({\rm i})},
& g_1, g_2, g_3 \in \mathfrak{g}^{({\rm i})},\\
- \langle [g_1,g_2],g_3 \rangle_{\mathfrak{g}^{(\l)}} \; (\sum_{\rm i} \l_{\rm i} v^{({\rm i})}),
& g_1, g_2, g_3 \in \mathfrak{g}^{(\l)},
\end{cases}
\end{equation}
\begin{align}
[u^{({\rm i})}, u^{({\rm j})},  x_\pm^{(\m)}]
&=& \pm \m_{{\rm i} {\rm j}} x_\pm^{(\m)}, \label{gen33} \\
\; [u^{({\rm i})}, x_+^{(\m)},  x_-^{(\m)}] &=& \sum_{\rm j} \m_{{\rm i} {\rm j}}  v^{({\rm j})}.
 \label{gen34}
\end{align}

It is quite remarkable that, starting out as a natural generalization
of a Cartan-Weyl algebras,
Cartan-Weyl 3-algebras as defined by
\eq{cw31}-\eq{cw34c} can be  constructed and classified
completely.

\section{Remarks on  Fuzzy $S^3$ in BLG Theory}

In the original construction of BLG,
the Lie 3-algebra  $\cA_4$ was employed.
The metric on $\cA_4$ is positive definite and the BLG
model defines a unitary quantum field theory.
The use of
$\cA_4$  was  motivated by the studies of
Basu and Harvey \cite{BH} whose main objective was to construct a
generalization of the Nahm equation for describing intersecting M-branes.
Employing a Lie 3-algebraic structure of $\cA_4$, the Basu-Harvey
equation  admits a solution whose cross section is given by a fuzzy
$S^3$ and describes
the puffing up of a system of multiple M2-branes into a M5-brane.

In general, one would like to employ more general metric Lie 3-algebras
with arbitrary higher ranks in the
BLG description of multiple M2-branes\footnote{
If one is interested only in the supersymmetric equations
of motion, it is possible to do without a metric \cite{nilsson}.}.
The choice of the
Lie 3-algebras should be such that the BLG theory or
generalization of the original construction, gives  a  unitary Quantum
Field Theory. The Lie 3-algebra should also allows different embedding of
$\cA_4$ in it in much the same way as one can find
fuzzy $S^2$  of different sizes in $SU(N)$.
Presumably the different $\cA_4$'s would then be
characterized by some kinds of Casimir of the $\cA_4$ algebra
which corresponds to
having different number of M2-branes puffing up to a single M5-brane
in the Basu-Harvey's fuzzy funnel solution of the multiple
M2-branes system.
However achieving these objectives turns out to be highly nontrivial.
For discussions about unitary BLG theory, see
\cite{LOR,GP,GG,Jind2,Jindn,d2d2,HMS}.
At present,  manifest unitary remains a major obstacle of the BLG theory.
To resolve it will require the use of a different kind of metric
Lie 3-algebras together with a novel ghost decoupling mechanism.
This is beyond the scope of this work and we will have nothing
more to say on this problem.

The problem of finding a fuzzy $S^3$ solution
has been considered recently for the ABJM theory \cite{ram}.
Surprisingly to the best
of our knowledge, the question of finding a fuzzy $S^3$
for the BLG theory in general has not been
considered before. We will examine  this issue
in details now.

It is
instructive to first recall the case of multiple D1-branes. There
the Nahm equation, the BPS
equation for the $U(N)$ non-abelian Born-Infeld theory of the
D1-strings, admits a fuzzy funnel whose transverse
cross section is a fuzzy $S^2$ and describes a bunch of D1-strings
puffing up into a D3-brane \cite{myers}, corresponding to a D1-D3 brane
intersection in string theory. Similarly, one would like to be able to
describe the M2-M5 branes
intersection as a fuzzy funnel solution of a BPS equation of
the multiple M2-branes theory.  In fact this was the original
motivation leading Basu and Harvey \cite{BH} to write down such a BPS
equation. It has been demonstrated in \cite{BL3} that the BLG
Lagrangian based on the algebra $\cA_4$ admits a fuzzy funnel solution whose
energy density scales as expected of a M2-M5 intersection (both
the massless  and massive cases).
However the problem has not been discussed for
BLG theory based on more general Lie 3-algebras, e.g. the
Lorentzian 3-algebra.

In general  a
fuzzy $S^3$ solution in the BLG theory
is described by having $X^\cP$, $\cP=1,2,3,4$ satisfying the $\cA_4$
algebra
\begin{equation} \label{A4}
[X^\cP, X^\cQ, X^\cR] = i \e^{\cP \cQ \cR}{}_\cS X^\cS.
\end{equation}
This describes a $SO(4)$ invariant distribution in the theory. In addition
one needs also a certain ``size'' condition so that one can be sure one is
describing a single fuzzy $S^3$. In the case of fuzzy $S^2$ for
D1-D3 intersection, the size condition is written down with respect
to the representation taken by the $X^\cP$'s which specifies the radius
of the fuzzy $S^2$. For fuzzy $S^3$, one can expect that one will
need to generalize the concept of a representation of Lie 3-algebras in order to
specify a suitable ``size'' condition. It is not known how to write
down arbitrary representations for a  general Lie 3-algebra. See \cite{wolf}
for some discussions.
However it is possible that for certain specific kinds of Lie 3-algebras,
for example, such as the Cartan-Weyl 3-algebras due to the Lie algebraic
structure they inherited.
In fact,
because of \eq{metric-cA} and \eq{metric3-eeh}, the invariant
metric of the Cartan-Weyl
3-algebra is precisely the same as the invariant metric
of the underlying Lie algebra.
Since the later can be immediately generalized to the Killing metric
which is defined for an arbitrary
representation of the Lie algebra, this can be extended to
the Cartan-Weyl 3-algebras immediately.
In the following, however, we will focus on the algebraic condition \eq{A4}
only  which is
independent of any representation issues.

Let us first  consider  the Lorentzian 3-algebra \eq{i1-5}. Let $X^\cP=
X^\cP{}_u u + X^\cP{}_v v+ X^\cP{}_{\hat{\a}} g^{\hat{\a}} $, where the Lie algebra
generators $g^{\hat{\a}}$ obeys $[g^{\hat{\a}}, g^{\hat{\b}}] = f^{{\hat{\a}}{\hat{\b}}}{}_{\hat{\g}} g^{\hat{\g}}$.
The equation \eq{A4} gives
\begin{align}
X^\cP{}_u &=& 0, \nonumber\\
 i \e^{\cP \cQ \cR}{}_\cS X^\cS{}_v &=& - X^\cP{}_{\hat{\a}} X^\cQ{}_{\hat{\b}} X^\cR{}_{\hat{\g}}
f^{{\hat{\a}}{\hat{\b}}{\hat{\g}}} ,
\\
 i
\e^{\cP \cQ \cR}{}_\cS X^\cS{}_{\hat{\d}}  &=&
X^\cP{}_{\hat{\a}} X^\cQ{}_{\hat{\b}} X^\cR{}_u f^{{\hat{\a}}{\hat{\b}}}{}_{\hat{\d}}
+ \mbox{($\cP,\cQ,\cR$ cyclic)}.\nonumber
\end{align}
Hence we obtain $X^\cS{}_u = X^\cS{}_{\hat{\d}} =X^\cS{}_v =0$ and
the Lorentzian BLG theory does not admit any fuzzy $S^3$ solution.

The above consideration can be generalized immediately to the general
Cartan-Weyl 3-algebra \eq{cw31}-\eq{cw34c}.
Let $X^\cP= X^\cP{}{}^I H_I + X^\cP{}_\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A} E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A} $.
The equation \eq{A4} gives
\begin{align}
i \e^{\cP \cQ \cR}{}_\cS X^\cS{}^L g_{LI} &=&
-\sum_{\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}+\b+\g=0} X^\cP{}_\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A} X^\cQ{}_\b X^\cR{}_\g g_I(\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A},\b) \nonumber\\
&&\quad + \sum_{\b} X^\cP{}^L X^\cQ{}_\b X^\cR{}_{-\b} \b_{LI}
+\mbox{($\cP,\cQ,\cR$
  cyclic)}, \label{XXX1} \\
i \e^{\cP \cQ \cR}{}_\cS X^\cS{}_\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A} &=& X^\cP{}^I X^\cQ{}^J X^\cR_\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A} \alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}_{IJ} \nonumber\\
&& \quad + \sum_\b X^\cP{}^I X^\cQ{}_\b X^\cR_{\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}-\b} g_I(\b, \alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}-\b)
+\mbox{($\cP,\cQ,\cR$
  cyclic)}.\;\;\;\;\;\; \label{XXX2}
\end{align}
Now given the general solution \eq{cond-ap}, \eq{cond-gI}, we can
multiply the equation \eq{XXX1} with $\ph^{({\mbox{{\suetterlin i}}})}{}^I$ and using the
properties \eq{cond6ap}, \eq{cond6p} to obtain that
\begin{equation}
X^\cS{}^I \ph^{({\mbox{{\suetterlin i}}})}{}_I=0, \quad \mbox{for all ${\mbox{{\suetterlin i}}}$ and $\cS$}.
\end{equation}
Substituting this into \eq{XXX2}, we than obtain  $X^\cS{}_\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}
=0$. This, substituting back into \eq{XXX1}, then implies that $X^\cS{}^I  =0$.
Therefore the Cartan-Weyl 3-algebra
does not admit  $\cA_4$ as a subalgebra.
We remark that this is quite different from the case of  Lie algebra where
any Lie algebra admits $SU(2)$ as a subalgebra.

The problem of finding a fuzzy $S^3$ solution
has also been considered in \cite{ram} for the ABJM
theory. For large value of the level $k$  where one can
trust the semi-classical analysis, it was found that
the ABJM theory admits only a fuzzy $S^2$ structure, rather than a
fuzzy $S^3$. This is the expected result since for generic $k$, the
geometry is $S^3/Z_k$ and the M-theory circle $S^1/Z_k$ becomes
zero in the $k \to \infty$ limit, reducing the system to the D2-D4
branes intersection in IIA string theory. Since the large $k$ limit is
needed in order to have a perturbative formulation of the ABJM theory,
finding a  fuzzy $S^3$ as the (semi-)classical geometry in the ABJM
theory is impossible.

For the BLG theory, there is a similar
problem. Presumably the level $k$  corresponds to some order of
orbifolding \cite{orbifold}
and a theory of multiple M2-branes in flat space is given by a
level one BLG theory with a certain choice of Lie 3-algebras.
It is not clear whether the fuzzy $S^3$ should emerge as  a classical
solution as in the original BLG theory \cite{BL3} or only  as a solution
of the full quantum system as in the ABJM theory.
Since the former situation is what happened in the original BLG
theory, it might be natural to expect that this to be also the case for
the BLG theories based on more general Lie 3-algebras.
Assuming that this is the case,
this no-go theorem
of finding a fuzzy $S^3$ solution in the Cartan-Weyl 3-algebras
then means that one need to consider more general class
of Lie 3-algebras for use
in the BLG theory. In the companion paper \cite{cat2},
a certain generalization of the Cartan-Weyl
3-algebras is suggested and we will show that
the no-go theorem can be bypassed easily in this class of Lie 3-algebras, giving
the hope of having fuzzy $S^3$ solution.

\section{Discussions}

In this paper we have generalized the notion of a Cartan-Weyl basis for a
Lie $3$-algebra. We have also shown that the consistency conditions
defining a Cartan-Weyl 3-algebra can be solved exactly, leading to a complete
classification of Cartan-Weyl 3-algebras. This is the main result of this paper.
It is natural to speculate that Cartan-Weyl 3-algebras may be useful
for describing
some  kinds of generalized symmetry. It will be
interesting to discover more of this.

We have mostly worked with metric Lie 3-algebras in this paper.
It is clear that
the concept of a Cartan-Weyl
basis can be similarly defined for  higher metric Lie $n$-algebras and
the existence of a Cartan-Weyl basis  is
equivalent to the requirement that the solutions of the
``eigenvalue equation''
\begin{equation}
[H_{I_1}, \cdots, H_{I_{n-1}}, Y^a] = \psi_{I_1 \cdots  I_{n-1}}{}^a{}_b Y^b
\end{equation}
are non-degenerate, and that the
restriction of the metric to the $H$'s is non-degenerate. As a result, a
Cartan-Weyl $n$-algebra is equipped with a number of
mutually commuting Cartan generators $H_I$
together with a number of step generators $E^\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}$
parametrized by a root space of non-degenerate $(n-1)$-forms $\alpha}      \def\da{{\dot\alpha}}  \def\dA{{\dot A}$.
As we have seen above, Cartan-Weyl
3-algebras have curiously built in them a structure of semisimple Lie
algebras. One can expect that Cartan-Weyl $n$-algebras will also
have built in them a certain special kind of Lie $(n-1)$-algebras.
It will be interesting to solve the
corresponding
consistency conditions and construct the Cartan-Weyl $n$-algebras
explicitly and find this out.
We speculate that these kinds of Lie $n$-algebras may
have a good chance to be of use in physics.

In order to understand which kind of Lie 3-algebras (or Lie
$n$-algebras in general) might appear in  physical
descriptions, it will be very helpful to understand how to
``integrate'' the infinitesimal transformations described by
Lie 3-algebras to
finite transformations since it is usually
much more clear how
the finite transformations should be constrained. One may call these
``Lie 3-group'' transformations.
There are many related interesting mathematical questions one may ask.
For example, Lie 3-group is certainly not a Lie group.
How is a Lie 3-group defined? It seems natural that  Lie
3-bracket may have its origin in a tertiary, perhaps nonassociative, product
structure of a Lie 3-group. Is it true?
Is there a Baker-Hausdorff formula?
While fascinating, there is nothing known in the literature
about how to think about a Lie 3-group.
Therefore we will have to look for other traits in order
to identify the desired type of Lie 3-algebras that could be relevant
for physical applications.

For applications in the
BLG models, the prospect of  Cartan-Weyl 3-algebras is not good
since the corresponding BLG
models are equivalent to ordinary Yang-Mills theories and so, not
surprisingly, they do not admit any fuzzy $S^3$ structure. Such BLG
models describe D-branes rather than multiple M2-branes.
We will argue in the paper  \cite{cat2} that the Lie 3-algebras of
interest
should satisfy a certain reduction condition. We will show that
this reduction condition leads to a class of metric Lie 3-algebras which
naturally generalizes the Cartan-Weyl 3-algebras  introduced
in this paper.
These  {\it generalized Cartan-Weyl 3-algebras}
have the same form \eq{ccw2}
of the Lie 3-brackets. However their Cartan subalgebra can be
non-abelian in general, i.e. the 3-brackets $[H_I,H_J,H_K] \neq 0$.
This modification is indeed a welcome one. We will explain in \cite{cat2}
how this modification may help with
getting fuzzy $S^3$ solutions  in the  corresponding BLG models.

\section*{Acknowledgements}

It is a pleasure to thank Douglas Smith for many useful discussions and
comments.
The research is supported by EPSRC and STFC.

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But, big thing...considering that Jesus was a b*stard (because his dad wasn't Joseph his lawful dad) then why is the Bible so concerned with paternity and genealogy and all those who begat whoms? Does it make any sense? If it does, I'd like to hear your thoughts.
Cuckoldry has always been a sore point with men . Perhaps this was an attempt to ease matters by using Joseph as an example of a man who took it well .
The genealogy bit comes from the fact that Mary was a descendant of King David, thus her son fulfilled the prophecy that a saviour would be born of David's line.
I never thought of that, Jacques - but, while we are on this prophesy-fulfilment track...did the prophesy say anything about a female descendant of King David called Mary getting impregnated by God ever come into the prophesy ?
Not as far as I remember, and I can't even remember where in the O.T. the prophecy is. I feel it is important to work out in which order things happened, or were writ. There seems to have been a lot of fiddling with time and events to fit. The travel to Bethlehem was one in question, they didn't have a census in those days, and even if they had, surely it would be taken where people were resident ? No, they had to go to Bethlehem, to fulfill that prophecy. Sorry PK. you touched a bit of a bee in my bonnet!!
Sorry about the bees in your bonnet, Jacques. I didn't mean to stir them up.
But, like you, I have often wondered about this Census thing in the Bible. I mean, why on earth did thousands of people have to trek back to the place they were born just to fill in in a bliddy census form? It's madness. Pure bureaucratic madness.
But that's Kings at the time of the Bible for you - Kings back then just didn't give a camel's f*rt about the ordinary people let alone worry about the effects of their royal dictats on the accommodation situation or hotel occupancy rates in Bethlehem. I'm very surprised that God didn't step in and use His omnipotence to try to find a decent vacant Travelodge room for the little infant Jesus' birth. It would have been the decent thing to do, even for an absentee father.


Your eyes have never seen such rich, luscious color. And at 1280x800, the resolution is well above normal HD standards. Of course, everything about the Tab is above normal.
We don't get much of a choice here in Russia, really only the Iconia, Xoom & the Samsung Galaxy, so far this Iconia is my favourite.
I got the Galaxy Tab over on Friday from a local BestBuy apparent they are not suppose to sell it until Monday. The screen has very wide viewing angle, I looked at the Ipad2 before I looked at the Galaxy Tab, and I couldn't tell the difference based on memory, but I'm sure Galaxy Tab is better. Though the screen is better than the Xoom which I also own.