= Differentiating left-invariant matrix fields
{title2=$[X_{B_1},X_{B_2}](Q)=Q[B_1,B_2]$}
On a <Matrix Lie group>, the <left-invariant vector field> associated to a <tangent vector> $B$ at the identity is $X_B(Q)=QB$. Its ambient derivative is $DX_B(Q)[H]=HB$. Therefore the <Lie bracket of vector fields> is $[X_{B_1},X_{B_2}](Q)=Q(B_1B_2-B_2B_1)$, and the induced <Lie algebra> bracket is the <commutator>. This fixes the sign for left invariance; right-invariant fields induce the opposite sign when evaluated with the same matrix identification.
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