I have a Lie algebra comprised of the generators $\{e_1,e_2,e_3,e_4\}$ for which the only non-zero commutators are
$ [e_4,e_2]=-i e_3 $ $ [e_4,e_3]= i e_2 $
(Excuse the physicist notation, for mathematicians the generators are actually ${ie_1,ie_2,ie_3,ie_4}$). Clearly, this is a solvable Lie Algebra ($[e_3,e_2]=0$).
My question is: Is there a simple way to refer to it without declaring its commutators? Like we can refer to $su(2)$ instead of writing out all of $su(2)$'s generators. I'd like to be able to do that for this algebra. If there's not a way to do this, I'd like to know that instead.