The first step in the construction is to "lift" the Lie bracket from the Lie algebra (where it is defined) to the tensor algebra (where it is not), so that one can coherently work with the Lie bracket of two tensors. The lifting is done as follows. First, recall that the bracket operation on a Lie algebra is a bilinear map formula_57 that is bilinear, skew-symmetric and satisfies the Jacobi identity. We wish to define a Lie bracket [-,-] that is a map formula_58 that is also bilinear, skew symmetric and obeys the Jacobi identity.