Electromagnetic charge matrix of electroweak gauge fields ($Q_{\mathrm{gauge}}=\left(\begin{array}{cccc}0& -i& 0& 0\\ i& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)$)

In the $(A^1,A^2,A^3,B)$ basis with gauge transformations $U=e^{i\alpha Q}$ and $Q=T_3+Y$, the non-Abelian connection transforms by conjugation. The Lie brackets $[T_3,T_1]=i{T}_2$ and $[T_3,T_2]=-i{T}_1$ give $\delta A^1=\alpha A^2$, $\delta A^2=-\alpha A^1$, with neutral $A^3,B$. Thus $\delta A=i\alpha Q_{\mathrm{gauge}}A$ gives the displayed Hermitian matrix. The fields $W^{\pm }=(A^1\mp i{A}^2)/\sqrt{2}$ transform as $e^{\pm i\alpha }{W}^{\pm }$, while both neutral combinations are uncharged. This separates the Adjoint representation on the gauge fields from the doublet charge matrix $\mathrm{diag}(1,0)$ on the Higgs field.