= Complex inclusion into quaternionic projective space
{title2=$i:\mathbb{CP}^n\hookrightarrow\mathbb{HP}^n,\quad i^*u=x^2$}
Coordinate inclusion of complex vectors sends a complex line to its quaternionic span. If two such vectors span the same quaternionic line, a nonzero complex coordinate forces their scalar ratio to be complex, so this map is injective. The pulled-back <quaternionic tautological line bundle> splits as $L\oplus\overline L$ with complex first <Chern classes> $-x,x$. Its top <Chern class>, and thus <Euler class>, is $-x^2$. With $u$ the negative Euler generator, $i^*u=x^2$ and $i^*(u^k)=x^{2k}$. The map on infinite projective spaces is injective on cohomology; a finite inclusion loses the degrees with $2k>n$.
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