Let $\frak{g}$ be a finite-dimensional complex nilpotent Lie algebra. What conditions on $\xi\in\frak{g}$ ensure the existence of a (canonical or non-canonical) surjective morphism $\frak{g}\rightarrow\frak{g}_{\xi}$ of complex Lie algebras? Also, what are some of the known decomposition theorems for finite-dimensional complex nilpotent Lie algebras?

A well-known decomposition theorem for nilpotent Lie algebras is the weight space decomposition, studied by R. Carles, L. J. Santharoubane and others (in the nilpotent case). Here $L$ is a finite-dimensional nilpotent Lie algebra over an algebraically closed field of characteristic zero, and $Der(L)$ its derivation algebra, $T$ a torus on $L$, that is, a commutative subalgebra of $Der(L)$ consisting of semisimple endomorphisms. This defines naturally a linear representation of $L$, and the elements of $T$ can be diagonalized simultaneously. Therefore we have the decomposition $$ L=\bigoplus_{\alpha \in T^{*}}L^{\alpha}, $$ where $L^{\alpha}$ is the set of $x\in L$ with $t(x)=\alpha (t)x$ for all $t\in T$. The equivalence class of a weight system is an invariant of $L$. It has been studied intensively for nilpotent Lie algebras, e.g., for the classification in low dimensions, and for the cohomology of nilpotent Lie algebras.

representationsof finite dimensional complex nilpotent Lie algebras. It states that every finite dimensional such representation is uniquely the direct sum of its generalised eigenspaces. $\endgroup$ – user91132 Jul 27 '12 at 8:04