Hermitian-matrix double cover of SO0(3,1) ($SL(2,\mathbb{C})/\{\pm I\}\cong S{O}_0(3,1)$)

On two-by-two Hermitian matrices, the determinant of $X=tI+x\cdot \sigma$ is $t^2-|x{|}^2$. The action $X\mapsto AX{A}^{†}$ 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.