Lie algebra generator

A set $S$ generates a Lie algebra $\mathfrak{g}$ when the smallest Lie subalgebra containing $S$ is $\mathfrak{g}$. Equivalently, every element is a finite linear combination of iterated Lie brackets of elements of $S$. A basis is always a generating set, but a generating set need not span $\mathfrak{g}$ by linear combinations alone: in the Heisenberg Lie algebra, $a,b$ generate because $[a,b]=c$, although their linear span has only two dimensions.

In a Matrix Lie group, physicists also call the matrices specifying infinitesimal transformations generators. Under the Hermitian quantum generator convention, a transformation has the form $I+i{\alpha }_a{t}_a+O(\Vert \alpha {\Vert }^2)$; the corresponding genuine Lie algebra elements are $i{t}_a$. A list of such generators may be a basis or a smaller generating set, so the distinction matters.