  Let $G$ be a compact, connected, and simply-connected Lie group, equipped
with a Lie group involution $\sigma_G$ and viewed as a $G$-space with the
conjugation action. In this paper, we present a description of the ring
structure of the (equivariant) $KR$-theory of $(G, \sigma_G)$ by drawing on
previous results on the module structure of the $KR$-theory and the ring
structure of the equivariant $K$-theory.
