We prove that the set of all Polish groups admitting a compatible complete left-invariant metric (called CLI) is coanalytic non-Borel as a subset of a standard Borel space of all Polish groups. As an application of this result, we show that there does not exist a weakly universal CLI group. This, in particular, answers in the negative a question of H.Becker.
"On Polish groups admitting a compatible complete left-invariant metric." J. Symbolic Logic 76 (2) 437 - 447, June 2011. https://doi.org/10.2178/jsl/1305810757