= Hermitian-matrix double cover of SO0(3,1)
{c}
{title2=$SL(2,\mathbb C)/\{\pm I\}\cong SO_0(3,1)$}
On two-by-two <Hermitian matrices>, the <determinant> of $X=tI+x\cdot\sigma$ is $t^2-|x|^2$. The action $X\mapsto AXA^\dagger$ preserves it for $A\in SL(2,\mathbb C)$. A kernel element is unitary and commutes with every Hermitian matrix, hence is $\pm I$. The derivative is injective and both real dimensions are six, giving the full proper orthochronous <Lorentz group> as the connected image.
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