Prove that for a 2 sphere in R3 the Lie bracket is the same as the cross product using the vector: X = (y,-x,0); Y = (0,z-y)
[X,Y] = JYX - JXY where the J's are the Jacobean matrices.
I computed JYX - JXY to get (-z,0,x). I computed (y,-x,0) ^ (0,z,-y) and obtained (xy,y2,yz) = (z,0,x)...