Continuity Criterion for the Restriction to the Commutator Subgroup of a Locally Bounded Finite-Dimensional Representation of a Connected Lie Group

We prove that a locally bounded finite-dimensional representation of a connected Lie group has a continuous restriction with respect to the original topology of the group to the commutator subgroup of the group if and only if the restriction of the representation to the center of a Levi subgroup is continuous with respect to the original topology of the group.