= Exterior-square double cover of SO0(3,2)
{title2=$Sp(4,\mathbb R)/\{\pm I\}\cong SO_0(3,2)$}
Fix a volume form on a four-dimensional real <symplectic vector space>. Wedge pairing on its <exterior square> has signature $(3,3)$. The <symplectic group> fixes the bivector inverse to the symplectic form. Its orthogonal complement is five-dimensional with signature $(3,2)$ after choosing the appropriate overall sign. The induced representation has kernel $\{\pm I\}$: identity action on the entire exterior square preserves every two-plane and thus every line, forcing a scalar transformation with square one. Its derivative is injective and both group dimensions are ten, so its image is the <identity component> of $SO(3,2)$.
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