We have the (P-local) commutative diagram:

[figure: Sp(n + 1); SU(2n + 2); \frac{SU(2n+2)}{Sp(n+1)}; Sp(l + 1); Σ^{4}SO(2n + 1); Σ^{4}SO(2l + 1); SU(2l + 2); \frac{SU(2l+2)}{Sp(l+1)}]