Steenrod powers on quaternionic projective space ($P^i(u^k)=\left(\genfrac{}{}{0ex}{}{2k}{i}\right)u^{k+i(p-1)/2}$)

Choose the degree-four generator with pullback $u\mapsto x^2$ under the complex inclusion into quaternionic projective space. On infinite complex projective space, the Cartan formula gives $P^i(x^{2k})=\left(\genfrac{}{}{0ex}{}{2k}{i}\right)x^{2k+i(p-1)}$. Injectivity of the infinite-space pullback proves the displayed formula for quaternionic projective space. Restriction to ${\mathbb{HP}}^n$ sets powers above $n$ to zero. Coefficients are reduced modulo the odd prime $p$; the exponent is an integer because $p-1$ is even.