= Euclidean reality structure on twistor space
{c}
{title2=$J^2=-1,\quad J(W)=W$}
Choose an antilinear quaternionic structure $J$ on $\mathbb C^4$. It induces a fixed-point-free involution on <projective twistor space>. A real Euclidean spacetime point is a $J$-invariant complex two-plane, equivalently a quaternionic line in $\mathbb H^2$. These form $\mathbb{HP}^1\cong S^4$, the <conformal compactification of Euclidean space>. Every projective twistor lies on the unique invariant line $\mathbb P\operatorname{span}_{\mathbb C}(Z,JZ)$, giving a fibration $\mathbb{CP}^3\to S^4$ with fiber $\mathbb{CP}^1$.
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