= Symmetric-matrix double cover of SO0(2,1)
{title2=$SL(2,\mathbb R)/\{\pm I\}\cong SO_0(2,1)$}
Congruence $X\mapsto AXA^T$ on real symmetric two-by-two <matrices> preserves the <quadratic form> $-\det X$ of signature $(2,1)$ when $\det A=1$. Its <kernel> is $\{\pm I\}$ and its derivative is injective: $aX+Xa^T=0$ for every symmetric $X$ implies $a=0$. Both <Lie groups> have dimension three, so the connected image is the full <identity component>. Also $A^TJA=(\det A)J$ identifies $Sp(2,\mathbb R)$ with $SL(2,\mathbb R)$.
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