The bi-invariant metric as pointed out by chombier is what I have in mind. I agree that a non-canonical metric may be the right one for some applications, but those are the exceptional ones. The bi-invariant metric (which has a simple, geometric meaning given by Euler's rotation theorem) is the right starting point for thinking about distances in this space.
(Your reference [2] supports this point of view: read "simplest" as "canonical". Your reference [1] claims five distinct bi-invariant metrics, but this is wrong. The argument given is that any metric related to a bi-invariant metric by a "positive continuous strictly increasing function" is itself bi-invariant, which is not true.)
(Your reference [2] supports this point of view: read "simplest" as "canonical". Your reference [1] claims five distinct bi-invariant metrics, but this is wrong. The argument given is that any metric related to a bi-invariant metric by a "positive continuous strictly increasing function" is itself bi-invariant, which is not true.)