  We use Richter's $2$-primary proof of Gray's conjecture to give a homotopy
decomposition of the fibre $\Omega^3S^{17}\{2\}$ of the $H$-space squaring map
on the triple loop space of the $17$-sphere. This induces a splitting of the
mod-$2$ homotopy groups $\pi_\ast(S^{17}; \mathbb{Z}/2\mathbb{Z})$ in terms of
the integral homotopy groups of the fibre of the double suspension
$E^2:S^{2n-1} \to \Omega^2S^{2n+1}$ and refines a result of Cohen and Selick,
who gave similar decompositions for $S^5$ and $S^9$. We relate these
decompositions to various Whitehead products in the homotopy groups of mod-$2$
Moore spaces and Stiefel manifolds to show that the Whitehead square $[i_{2n},
i_{2n}]$ of the inclusion of the bottom cell of the Moore space $P^{2n+1}(2)$
is divisible by $2$ if and only if $2n=2, 4, 8$ or $16$.
