Is every 3D Lie group with bi-invariant metric a space form? I seem to remember to have read somewhere that, "in dimension three, every bi-invariant metric on a Lie group has constant curvature". However, I am not able to find a confirmation of this in any reference I know.
Can anybody please confirm/disprove such statement?

The answer can be found in Milnor's paper
Curvatures of Left Invariant Metrics on Lie Groups.
A connected Lie group G admits a bi-invariant metric if
and only if is isomorphic to the Cartesian product of a compact group and a vector space $\Bbb R^m$. In this case, it is a symmetric space. In dimension $3$, there are not so many possibilities and we can check this directly.
