= Torsion contractions of a left-parallel Lie-group connection
{title2=$T_a{}^a{}_c=0,\quad T_a{}^b{}_cT_b{}^a{}_d=K_{cd}$}
Declare a global frame of <left-invariant vector fields> parallel on a <semisimple Lie group>. This <affine connection> is flat and preserves a left-invariant <metric tensor> because its frame components are constant. If $[e_a,e_c]=C_a{}^b{}_ce_b$, its <torsion tensor> is $T_a{}^b{}_c=-C_a{}^b{}_c$. For fixed $c$, the matrix with entries $T_a{}^b{}_c$ is $\operatorname{ad}e_c$. Its trace vanishes because a semisimple <Lie algebra> is perfect and adjoints of brackets are commutators. The trace of the product for $c,d$ is the <Killing form> $K_{cd}$, proving both contractions. This flat metric connection differs from the curved torsion-free <Levi-Civita connection> of the Killing metric.
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