Quaternionic tautological line bundle ($\mathcal{H}\to {\mathbb{HP}}^n$)

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 $-c_1(L^{\ast }{)}^2$.