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?



Lie bracket of algebraic vector fields
In [an old blog post](http://qchu.wordpress.com/2011/02/26/the-quaternions-and-lie-algebras-i/) I give a definition of the Lie bracket of two derivations (of a ring, not necessarily commutative, in which $2$ is invertible) that doesn't require limits and doesn't require that you verify that the commutator of two derivations is a derivation; the whole thing naturally falls out of the careful use of nilpotents. Since the Lie bracket is local, for varieties it suffices to piece together the story of that post on open affines. 


