Remark

Let G and $G'$ be two connected compact Lie groups, and let $e=$ $(T,B,(U_{\alpha}))$ and $e'= (T',B',(U'_{\alpha}))$ be framings of G and $G'$, respectively. Let X be the set of isomorphisms from G to $G'$ that take $e$ to $e'$. The map $f \rightarrow D^*(f)$ (resp. $D_*(f)$) is a bijection from X to the set of isomorphisms from $D^*(G',T')$ to $D^*(G,T)$ (resp. $D_*(G,T)$ to $D_*(G',T')$) that map $B'$ to B (resp. B to $B'$). Indeed, this follows immediately from Prop. 15 and Lemma 3.