= Quaternionic structure on a complex vector space
{title2=$J^2=-1$}
An antilinear endomorphism $J$ with square $-1$ equips a <complex vector space> with a quaternionic structure: complex multiplication supplies the quaternionic generator $i$ and $J$ supplies $j$, satisfying $Ji=-iJ$. Such a space has even complex dimension. There is no invariant complex line, because $Jv=cv$ would imply $-v=|c|^2v$. In dimension four the invariant complex two-planes are quaternionic lines, forming $\mathbb{HP}^1\cong S^4$. This is the <Euclidean reality structure on twistor space> used in <twistor theory>.
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