Since we are dealing with symmetric spaces, in particular with compact semi-simple Lie groups, we already know that the space can be equipped with a bi-invariant metric. As you mentioned before, we know that compact semi-simple Lie groups with a bi-invariant metric form a symmetric space. However, the results on which you base your assumption were derived by heavily depending on the symmetric space structure. This seems to be a total loop for me (in the methodological sense!).

If you are referring to the general question below, I would say that I do not see the point in proving something like that. This is not meant as an offence, but I actually do not understand where this question is supposed to be going. All considerations above were made for symmetric spaces, that require the compact Lie Group to be already equipped with a bi-invaraint metric.

I know this is not quite an answer, but I don't know how to comment, so if someone of the moderators feels this should be converted in a comment, I'd not necessarily oppose to doing that.