A CW-group $G$ is a topological group with a CW structure such that both the multiplication and the inversion are cellular. It is known that $O(n)$, $U(n)$, and $Sp(n)$ can be given CW-group structures: for the CW structures given in Steenrod and Epstein, Cohomology Operations, Chap. IV, it is not hard to check that the relevant maps are cellular.
$SU(n)$ can be given a CW structure, via the homeomorphism $SU(n) \approx G_{\mathbb C}(n,1) := U(n)/U(1)$. This is what was done in Cohomology Operations, as well as "On the cell structures of $SU(n)$ and $Sp(n)$," but my understanding is that this does not make $SU(n)$ a CW-group.
Can $SU(n)$ be given a CW-group structure? Is there any theorem addressing the question as to when a topological group can be given a CW-group structure?