  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 $p\ne 3$ 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 $G\simeq_{(p)}H\times G/H$,
which yields a projection $G_{(p)}\to H_{(p)}$. We show how much this
projection preserves the higher homotopy associativity.
