Let $X$ be an algebraic variety (over any field). The definition of the Lie bracket of two vector fields on $X$ (i.e. sections of the tangent sheaf) which I know characterizes vector fields as derivations of the structure sheaf, and then one checks that the commutator of two derivations is a derivation. My first question is whether this definition works when $X$ is not smooth.

I am more concerned about my second question if the answer to the first is no, but I am interested regardless: is there an "intrinsic" definition of the Lie bracket which does not require us think of vector fields as operating on functions?

lesselegant than the other definition. – Theo Johnson-Freyd Apr 17 '12 at 3:56