As Yves Cornelier already said: Your presumed statement is wrong.

Any connected, linear, solvable Lie group over the reals is the semi-direct product of a compact abelian subgroup and a simply connected normal subgroup. (This holds more general for algebraic, connected, solvable lie groups ober a field of characteristic 0, what can be found in Chevalley's "Théorie des groupes de Lie")

This is in general false for non-linear Lie groups, what explains Yves Cornulier's counter example as the quotient of the heisenberg group with it's central discrete cyclic subgroup is not linear.

(The non-linearity of this group is e.g. proved in "The Structure of Compact Groups: A Primer for Students, a Handbook for the Expert" of Hofmann and Morris on page 169.)

As Yves Cornulier already said: Your presumed statement is wrong.

Any connected, linear, solvable Lie group over the reals is the semi-direct product of a compact abelian subgroup and a simply connected normal subgroup. (This holds more general for algebraic, connected, solvable lie groups ober a field of characteristic 0, what can be found in Chevalley's "Théorie des groupes de Lie")

This is in general false for non-linear Lie groups, what explains Yves Cornulier's counter example as the quotient of the heisenberg group with it's central discrete cyclic subgroup is not linear.

(The non-linearity of this group is e.g. proved in "The Structure of Compact Groups: A Primer for Students, a Handbook for the Expert" of Hofmann and Morris on page 169.)