= Solution
The <special linear group> acts on the <twistor correspondence> by $g\cdot(\ell,W)=(g\ell,gW)$. Since an invertible linear map preserves inclusion, this is again an incident flag. The two projections commute with that action:
$$
\boxed{\nu(g\cdot(\ell,W))=g\cdot\nu(\ell,W),\qquad\mu(g\cdot(\ell,W))=g\cdot\mu(\ell,W).}
$$
This is precisely <equivariance>: projecting a transformed flag gives the same result as transforming its projection. In particular $g$ sends the <twistor line> of a spacetime point to the line of the transformed point and sends the <alpha-plane> of a twistor to that of its transformed twistor. The incidence geometry is preserved, even when a transformation takes a point out of the chosen affine Minkowski chart.
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