SL(2,C) spin cover of the Lorentz group

Statement

Under the Hermitian matrix model, the action

X\longmapsto AXA^\dagger

defines a surjective real Lie group homomorphism

\rho:SL(2,\mathbb C)_{\mathbb R}\longrightarrow SO^+(1,3)

whose kernel is $\{\pm I\}$ . Hence $\rho$ is a two-sheeted covering and realizes

SL(2,\mathbb C)_{\mathbb R}\cong\operatorname{Spin}^+(1,3).

Kernel, image, and differential

If $AXA^\dagger=X$ for every Hermitian $X$ , then $A$ commutes with the resulting full matrix algebra and is scalar; determinant one leaves $A=\pm I$ . The image lies in the identity component because $SL(2,\mathbb C)$ is connected. Its differential is an injective map between real Lie algebras of dimension $6$ , so the image is open; connectedness and the standard structure of $SO^+(1,3)$ give surjectivity.

Differentiating yields the real Lie algebra isomorphism

\mathfrak{sl}_2(\mathbb C)_{\mathbb R} \xrightarrow{\;\sim\;} \mathfrak{so}(1,3), \qquad X\longmapsto\bigl(H\mapsto XH+HX^\dagger\bigr).

This is not an isomorphism of complex Lie algebras because $\mathfrak{so}(1,3)$ is here a real Lie algebra.

References

Roger Penrose and Wolfgang Rindler, Spinors and Space-Time, Vol. 1, Cambridge University Press, 1984, §§1.2–1.3. Publisher record.
H. Blaine Lawson and Marie-Louise Michelsohn, Spin Geometry, Princeton University Press, 1989, Chapter II, §5. Publisher record.