Unique normal subpair below a Brauer pair

Partition the primitive central block decomposition of $1$ in $k{C}_G(Q)$ into its $P$-orbits. The non-singleton orbit sums map to zero. The remaining blocks are individually $P$-stable. Their relative images are orthogonal central idempotents whose sum is one: multiplicativity preserves idempotence and orthogonality. For centrality, lift an arbitrary target element through the surjective relative map; its lift commutes with each central source block, so their images commute.

Multiply this sum by primitive central $e$. Each product is either $0$ or $e$, since a nontrivial product would split $e$; at least one equals $e$ because the sum of products is $e\mathrm{\ne }0$, and at most one can equal $e$ by orthogonality. Its preimage block $f$ is $P$-stable and satisfies the required equation. Every normal subpair must be one of these $P$-stable blocks by [F1], proving uniqueness as well as existence.