= Identity augmentation in quantum controllability
{title2=$\mathfrak g+\mathbb R iI=\mathfrak u(N),\quad\operatorname{Tr}\mathfrak g\ne0\Rightarrow\mathfrak g=\mathfrak u(N)$}
For $N\ge2$, suppose a skew-Hermitian <Lie algebra> becomes all of $\mathfrak u(N)$ after adding the scalar identity direction. Scalar matrices commute, so $[\mathfrak g,\mathfrak g]=[\mathfrak u(N),\mathfrak u(N)]=\mathfrak{su}(N)$ is contained in $\mathfrak g$. If $\mathfrak g$ contains any nonzero-trace generator, subtracting its traceless part supplies $iI$, and $\mathfrak g$ was already all of $\mathfrak u(N)$. This permits using identity-augmented commutator calculations without assuming an additional physical identity control.
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