= Solution
Use the <Clifford algebra> to write $\gamma^\nu\gamma^\mu=2\eta^{\mu\nu}I_4-\gamma^\mu\gamma^\nu$. The <Lorentz generator from gamma-matrix commutators> is therefore
$$
S^{\mu\nu}=\frac14[\gamma^\mu,\gamma^\nu]=\frac12\gamma^\mu\gamma^\nu-\frac12\eta^{\mu\nu}I_4,
\qquad
\boxed{A=\frac12,\quad B=-\frac12.}
$$
This also checks the diagonal case $\mu=\nu$, where $S^{\mu\mu}=0$ because $(\gamma^\mu)^2=\eta^{\mu\mu}I_4$. The metric term in the printed expression is understood to multiply the <identity matrix>. The generators are antisymmetric in their two spacetime indices.
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