= Lie algebra generator
{c}
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 $I+i\alpha_a t_a+O(\|\alpha\|^2)$; the corresponding genuine <Lie algebra> elements are $it_a$. A list of such generators may be a <basis> or a smaller <generating set>, so the distinction matters.
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