In the standard model (omitting the QCD part), we start off with the set of generators
$T_1$, $T_2$, $T_3$, $Y$
for the four-parametric gauge group $SU(2)_L \times U(1)_Y$.
We then define a new generator $Q= T_3+Y$ and make the transition to the four-parametric gauge group $SU(2)_? \times U(1)_Q$.
What are, aside from $Q$, the new generators for this "new" gauge group?
$?$ , $?$ , $?$ , $Q$
Do we still use the $T$'s we used in $SU(2)_L$? That means the left factor in the group product is still the same as before the symmetry breaking?
My motivation for asking is the observation that in $SU(2)_L \times U(1)_Y$, the four generators are orthogonal and a basis for the space of all complex self-adjoint matrices.
The set of $T_1$, $T_2$, $T_3$, $Q$, while still a basis, is however not orthogonal, since
$( T_3| Q )$=$(T_3|T_3+Y)$=$(T_3|T_3) \neq 0$
It would seem that we would probably want to preserve that orthogonality property and thus not use $T_3$ as a generator after symmetry breaking.