We prove a closed form formula for the stochastic exponential of a matrix-valued semimartingale under the assumption that various commutativity conditions are fulfilled. This extends a corresponding result for continuous semimartingales by Yan in [9] to semimartingales with jump parts. We give three examples of a semimartingale in a Lie group setting for which the commutativity conditions are easily verified and explicitly compute their stochastic exponential.