I am putting together an exposition on Lie theory; maths research is not my day job, let alone real maths history, so apologies in advance for any ignorance shown by these questions.

I am attempting a translation (with explanatory notes) of B. L. van der Waerden's 1932 papper "Stetigkeitssätze für halbeinfache Liesche Gruppen" (Continuity Theorem for Semisimple Lie Groups), see here. In this paper, an arbitrary sequence $\left\{C_{\nu, \mu}\right\}$ of elements $C_{\nu, \mu}$ belonging to the identity-connected component of a compact Lie group is considered, and a convergent subsequence $\left\{C_\nu\right\}$ picked out.

Clearly there is no problem with any of this, but a couple of details I would fill in are: (1) in compact connected Lie groups the exponential map is surjective, so that sequences thereof are equivalent to sequences in the Lie algebra, whence (2) Bolzano-Weierstrass is readily applied.

So I'm curious as to whether (1) van der Waerden is just assuming these details would be known to and filled in by one of his readers or (2) whether instead he had a slightly different concept of "compact lie group" to our modern one, perhaps something like a group wherein an obvious generalisation to Bolzano-Weierstrass applies. What would someone in 1932 understand by the idea of a compact group, or indeed of compactness in general; would it have been exactly the same as what we would understand today?