Rotation Group SO(3) - Lie Algebra

Lie Algebra

Since SO(3) is a Lie subgroup of the general linear group GL(3), its Lie algebra can be identified with a Lie subalgebra of gl(3), the algebra of 3×3 matrices with the commutator given by

The condition that a matrix A belong to SO(3) is that

(*)

If A(t) is a one-parameter subgroup of SO(3) parametrised by t, then differentiating (*) with respect to t gives

and so the Lie algebra so(3) consists of all skew-symmetric 3×3 matrices.