Question: I understand the proper velocity is the velocity as measured by the traveler, and relative velocity as measured by the observer.

Relative velocity is limited to $c$, but is that also true of proper velocity? Theoretically, of course, assuming a sufficient energy source was available. And what would be the effects on the traveler if it was possible for proper velocity to exceed $c$?

Answer: 
1. The commutation relations (1) form the real Lie algebra $so(3,\mathbb{R})$ in the physics conventions, where the Lie algebra elements are chosen Hermitian.

In contrast in the mathematics convention, where the Lie algebra elements are chosen anti-Hermitian, there is no explicit imaginary unit $i$ in the commutation relation (1) for $so(3,\mathbb{R})$. In other words, the [structure constants](https://en.wikipedia.org/wiki/Structure_constants) are real. This explains why it's a real Lie algebra. See also my related Phys.SE answer [here](https://physics.stackexchange.com/a/365549/2451).

The complexification is isomorphic to $so(3,\mathbb{C})$.
2. The commutation relations (2) form the real Lie algebra $sl(2,\mathbb{R})\cong so(1,2;\mathbb{R})$ in the mathematics convention.
See also [this](https://physics.stackexchange.com/q/68333/2451) related Phys.SE post.

Their complexification is isomorphic to $so(3,\mathbb{C})$.

The above is a good example why it is important to distinguish between real and complex Lie algebras.
A reasonably simple way to disentangle this is to start from the group. Surely a rotation by an angle $\theta$ about $\hat z$ would be represented by the real matrix
\begin{align}
R\_z(\theta)&= 
\left(\begin{array}{ccc}
\cos\theta & \sin\theta & 0 \\
-\sin\theta &\cos\theta &0 \\
0&0&1\end{array}\right)\, \tag{1}
\end{align}
etc. Note that of course (1) is NOT a diagonal matrix with complex entries, but a real matrix which cannot be made diagonal without introducing complex numbers.

The generator of infinitesimal rotation (defined without the "i" as is traditional in physics)
\begin{align}
\hat {\mathbb{L}}\_z=\frac{d}{d\theta}R\_z\bigl\vert\_{\theta=0}
\end{align}
would be the *real* antisymmetric matrix
\begin{align}
\hat {\mathbb{L}}\_z = 
\left(\begin{array}{ccc}
0 & 1 & 0 \\
-1 &0 &0 \\
0&0&0\end{array}\right)\, \tag{2}
\end{align}
and NOT hermitian.

You see how the physics convention would differ as the generators are defined with an $i$ in it:
\begin{align}
\hat {{L}}\_z=-i\frac{d}{d\theta}R\_z\bigl\vert\_{\theta=0}\, .
\end{align}

The introduction complex numbers is required at some point because of the insistence on using diagonal operators. The eigenvectors of (2) are complex combination of the basis vectors $\hat{\boldsymbol{e}}\_{x,y,z}$.

The factor of "$i$" is of course not an issue if you are dealing with matrices with complex entries, such as $SU(2)$.

In dealing with real form and complex extensions, the mathematics way of doing things is less confusing although not familiar to physics. The only math/phys. book I know who consistently follows the math convention is

> 
> Cornwell, J.F., 1984. Group theory in physics. 2 (1984). Acad. Press.
> 
> 
> 

If you deal with compact groups, then one can complexify and decomplexify without second thoughts. If you are dealing with non-compact groups (*v.g.* Lorentz), then one has to be careful as representations that are irreducible under the reals may become reducible over the complex (*v.g.* Lorentz again: if you're not allowed to take the combo $K\pm iL$ then the adjoint is irreducible and does not break into $\mathfrak{su}(2)\oplus \mathfrak{su}(2)$).