= Exterior-square double cover of SO(3,3)
{title2=$SL(4,\mathbb R)/\{\pm I\}\cong SO_0(3,3)$}
A fixed volume form makes wedge pairing on $\Lambda^2\mathbb R^4$ a symmetric form of signature $(3,3)$. The exterior-square action of <special linear group> <matrices> preserves it. Its kernel is $\{\pm I\}$: a transformation acting trivially on every bivector preserves every two-plane and hence every line, so is scalar, with scalar square one. The derivative is injective and both groups have dimension fifteen, so the connected image is the full <identity component>. This is a concrete real spin-group cover.
Back to article page
