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.