Suppose $A$ and $V$ are the generators of some Lie algebra $\mathfrak g$. Furthermore, the $V$'s generate a subalgebra $\mathfrak v$ of $\mathfrak g$.
The $A$'s and $V$'s commute as follows:
$$ [V,A]=sA $$
for some matrix $s$. This property is convenient, because then the exponentiated Lie group elements braid nicely:
$$ e^{V}e^{A} = e^{(e^{s})A}e^{V}. $$
Does the subalgebra $\mathfrak v$ has any special name due to the above property?
As an example, say $V_i$ and $A_i$ (with $i=1,2,3$) are the six generators of $\mathfrak{so}$(4); furthermore, $V_i$ are the generators of the subalgebra $\mathfrak{so}(3)$. Then it just happens to be the case that:
$$ [V_i,A_j]=\epsilon_{ijk}A_k. $$
I'm not sure about special names... why do you need them? You are describing the standard current algebras of QFT. The Vs are in the subalgebra and the As are in the coset space.
In your particular example, the Vs are the rotation group generators, so the "name" one describes the axials A with is "they transform as a vector of the rotation group". Your group composition braiding identity is merely the adjoint action of rotations on axial-generated group elements, AdR exp(A).
If you endow a grading (+) on the Vs in the subalgebra and (-) on the As, then you get the reminder from the 3 types of commutation relations in the algebra that that is graded, that is, the gradings (parity in physics) of the left- and right-hand sides match, upon multiplicative composition of the individual gradings.