= Lorentz reality condition on chiral generators
{c}
{title2=$A_i^\dagger=B_i$}
For Hermitian rotations $J_i$ and boosts $K_i$, the complex combinations $A_i=(J_i+iK_i)/2$ and $B_i=(J_i-iK_i)/2$ are adjoints of one another. Although their complex brackets split into two commuting $\mathfrak{sl}_2$ factors, they are not two independent Hermitian compact rotation algebras. The real Lorentz <spin> group is $SL(2,\mathbb C)$; $SU(2)\times SU(2)$ is a different compact real form.
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