= Quaternionic tautological line bundle
{title2=$\mathcal H\to\mathbb{HP}^n$}
Its fibre over a point of <quaternionic projective space> is the quaternionic line represented by that point. As a real vector bundle it has rank four, and its unit <sphere bundle> is $S^{4n+3}\to\mathbb{HP}^n$. Right multiplication by a chosen imaginary unit equips it with complex rank two. Under the <complex inclusion into quaternionic projective space>, it restricts to $L\oplus\overline L$, where $L$ is the complex <tautological bundle>. Thus its <Euler class> restricts to $-c_1(L^*)^2$.
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