Lie algebra generated by a set

Question

Let L be a Lie algebra and $\{e_i,h_i,f_i; i=1 \ldots l\}$ be a basis of L. Is it true that the subalgebra of L generated by $\{e_i,h_i,f_i; i=1 \ldots l\}$ equals whole of L? What is the form of the elements of subalgebra generated by $\{e_i,h_i,f_i; i=1 \ldots l\}$? I am not getting form a subalgebra generated by a finite subset of a Lie algebra L?

Answer 1

The above set of generators is the classical set of generators of a complex simple Lie algebra of rank $l$. This is usually called the Chevalley generators' set, which satisfying some relations related to a Cartan matrix, you can find them for instance in H. Samelson - Notes on Lie algebras, p.73 or N. Jacobson - Lie algebras, p.125. The generating space of a set S is the sum $\sum_{i\geq0}S_i$, where $S_0=S$ and $S_i=[S_{0},S_{i-1}]$, $i\geq1$, which is the least subalgebra containing $S$.