Complex matrix Lie algebra

Let be a Matrix Lie group, and denote the set of curves in intersecting identity at zero, i.e.

The matrix Lie algebra is given by

i.e. the tangent space at identity, with the Lie bracket given by the Linear commutator:

Proof of Lie algebra

By properties of the Linear commutator, is alternating bilinear and satisfies the Jacobi identity. All that remains to show is that is closed under this operation. To this end let so that and . Now define a curve by . Since is a vector space and the Tangent space at any point in a vector space is the vector space, . From the product rule

and

it follows that and thus

hence is closed under the Lie bracket.

Basis

A good choice of basis uses coördinate lines in a chart containing the identity matrix. Let be a -dimensional coördinate chart with . Then