= Every proper orthochronous Lorentz transformation is an exponential
{title2=$L=\exp\Omega$}
Use the <Lorentz spinor double cover> $SL(2,\mathbb C)\to SO^+(1,3)$. For a lift with distinct eigenvalues $\lambda,\lambda^{-1}$, diagonalize and choose logarithms $z,-z$; this gives a traceless matrix logarithm. For a repeated eigenvalue, choose the sign of the lift so it is $I+N$, where $N^2=0$, and use $\log(I+N)=N$. The two lifts have the same Lorentz image. Exponentiating the differential of the covering map therefore expresses every proper orthochronous transformation as the exponential of a real element of the <Lie algebra of the Lorentz group>. This argument does not assert that every element of $SL(2,\mathbb C)$ itself has a traceless logarithm.
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