 Solution

Projection onto the subring in part (b), along the two-sided ideal, is the Brauer homomorphism:It is a unital algebra homomorphism because it is the quotient by that ideal followed by the displayed subring identification.

Put $N=N_G(D)$. The projection is $N$-equivariant, since $N$ preserves the centralizer. Its restricted kernel isEquivalently it is spanned by the sums of the $N$-conjugation orbits on $G$ disjoint from $C_G(D)$. Normality of $D$ in $N$ ensures that an $N$-orbit either lies wholly inside that centralizer or wholly outside it.