 Exponential product defect identity

For finite matrices, differentiate the ordered product to obtain ${\Phi }^{\prime }-(A+B)\Phi =[e^{tA},B]e^{tB}$. Multiplying the error by $e^{-t(A+B)}$ and integrating gives $\Phi (t)-e^{t(A+B)}={\int }_0^t{e}^{(t-s)(A+B)}[e^{sA},B]e^{sB}ds$. This exact variation-of-constants formula identifies noncommutativity as the source of Lie-Trotter splitting error.