Higher homotopy associativity in the Harris decomposition of Lie groups

Let $(G,H)=(SU(2n+1),SO(2n+1)),(SU(2n),Sp(n)),(SO(2n),SO(2n-1)),(E_6,F_4),(Spin(8),G_2)$, and let $p$ be any prime $\ge 5$ for $(G,H)=(E_6,F_4)$, any prime $p\mathrm{\ne }3$ for $(G,H)=(Spin(8),G_2)$, and any odd prime otherwise. The classical result of Harris on the relation between the homotopy groups of $G$ and $H$ is reinterpreted as a $p$-local homotopy equivalence $G{\simeq }_{(p)}H\times G\mathrm{/}H$, which yields a projection $G_{(p)}\to H_{(p)}$. We show how much this projection preserves the higher homotopy associativity.