Question: I have a nilpotent lie group $N$ with upper central series
$$1 = N\_0 \triangleleft N\_1 \triangleleft \dots \triangleleft N\_k = N$$
which induces the filtration $$0 = \mathfrak{n}\_0 \subset \mathfrak{n}\_1 \subset \dots \subset \mathfrak{n}\_k = \mathfrak{n}$$
of the Lie algebra $\mathfrak{n}$. 

For convenience, I've defined the *level* of a vector $\xi \in \mathfrak{n}$ to be the smallest $i$ such that $\xi \in \mathfrak{n}\_i$. I assume this concept has a standardized name, either in the context of filtrations or in the context of nilpotent Lie algebras. What would this standardized name be?

Answer: 
I doubt very much that a "standardized" name for the concept exists in the context of nilpotent Lie algebras, to judge by a quick look at older books by Bourbaki (Chapter 1,
1960, *Groupes et algebres de Lie*), Jacobson, Dixmier. I'm less familiar with terminology in the theory of Lie rings and abstract nilpotent groups. Once the terms of the upper (or ascending) central series are labelled as something like $\mathfrak{n}\_i$, it's typical just to refer to an element $x$ lying in $\mathfrak{n}\_i$ but not in the previous term. Perhaps somewhere in the literature a term like *level* or *height* or whatever might get used to refer to the index $i$ here, but such terms tend to be used for other purposes in Lie theory (I've even seen a reference to "level of a nilpotent Lie algebra" in the context of varieties of Lie algebras).
Probably *filtration level* would be safe when an ascending filtration is fixed.
I am not sure that there is a standard answer, so you will need to define it. But in profinite group theory we tend to call it degree, (if you have a pro-$p$ group and nice enough filtration, you can associate a graded Lie algebra and elements in the group map to homogenous elements in the Lie algebra, so degree makes sense), although, we usually deal with descending filtrations.