Continuity criteria for locally bounded homomorphisms of certain Lie groups (Q6608064)

In the sequel to his previous articles on the problems related to continuity of locally bounded homomorphisms of connected Lie groups [Proc. Jangjeon Math. Soc. 26, No. 2, 221--225 (2023; Zbl 1524.22012); Russ. J. Math. Phys. 30, No. 1, 126--127 (2023; Zbl 1530.22006); Adv. Stud. Contemp. Math., Kyungshang 31, No. 4, 501--504 (2021; Zbl 1492.22016)], the author here proves the following:\N\NLet \(G\) be a connected Lie group whose commutator subgroup is closed and \(\alpha\) be a locally bounded homomorphism from \(G\) into a Lie group. Then there is a supplementary subgroup \(Z\) of \(G\) such that \(G=G' Z\) and \(\alpha\) is continuous on \(G\) iff it is continuous on \(Z\).