= Left-right double cover of SO0(2,2)
{title2=$SL(2,\mathbb R)^2/\{(I,I),(-I,-I)\}\cong SO_0(2,2)$}
The action $(A,B):M\mapsto AMB^{-1}$ preserves the <quadratic form> $-\det M$ on real two-by-two matrices. Its <kernel> consists of the simultaneous scalar pairs $(I,I)$ and $(-I,-I)$. The differential $(a,b):M\mapsto aM-Mb$ is injective on the two traceless <Lie algebras>, and both source and target have dimension six. Its image is therefore open and, since the source is connected, equals the <identity component of a Lie group> $SO_0(2,2)$. This gives a two-sheeted <covering map> onto the identity component, not onto the disconnected full <special orthogonal group> $SO(2,2)$.
Back to article page
