Suppose formula_176 is a real Lie group with Lie algebra formula_1. Following the modern approach, we may identify formula_1 with the space of left-invariant vector fields (i.e., first-order left-invariant differential operators). Specifically, if we initially think of formula_1 as the tangent space to formula_176 at the identity, then each vector in formula_1 has a unique left-invariant extension. We then identify the vector in the tangent space with the associated left-invariant vector field. Now, the commutator (as differential operators) of two left-invariant vector fields is again a vector field and again left-invariant. We can then define the bracket operation on formula_1 as the commutator on the associated left-invariant vector fields. This definition agrees with any other standard definition of the bracket structure on the Lie algebra of a Lie group.