The structure of locally bounded finite-dimensional representations of connected locally compact groups

An analogue of a Lie theorem is obtained for (not necessarily continuous) finite-dimensional representations of soluble finite-dimensional locally compact groups with connected quotient group by the centre. As a corollary, the following automatic continuity proposition is obtained for locally bounded finite-dimensional representations of connected locally compact groups: if $G$ is a connected locally compact group, $N$ is a compact normal subgroup of $G$ such that the quotient group $G/N$ is a Lie group, $N_0$ is the connected identity component in $N$, $H$ is the family of elements of $G$ commuting with every element of $N_0$, and $\pi$ is a (not necessarily continuous) locally bounded finite-dimensional representation of $G$, then $\pi$ is continuous on the commutator subgroup of $H$ (in the intrinsic topology of the smallest analytic subgroup of $G$ containing this commutator subgroup).
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