Exterior-square double cover of SO(3,3) ($SL(4,\mathbb{R})/\{\pm I\}\cong S{O}_0(3,3)$)

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.