I have seen two definitions of the Lie Bracket for a Riemannian manifold $(M,g)$.
One is this : $[X,Y] = D_X Y - D_Y X$, where $D$ stands for covariant differentiation. When written out, this seems to involve the Christoffel symbols. (This is in Thorpe's Elementary Topics in Differential Geometry.)
The other is this: $[X,Y] = XY - YX$ with $(XY - YX)(f) = (X(Y(f)) - Y(X(f))$. (And the thing about differentiation of Y along the flow of X.) (For instance, in Warner's Differential Manifolds.)
So my confusion is this: one expression seems to involve the metric, the other does not. How can I reconcile this?