Unbroken neutral gauge group of a zero-hypercharge real triplet ($SU(2)\times U(1{)}_Y\to U(1{)}_{T^3}\times U(1{)}_Y$)

A real scalar triplet of weak hypercharge zero with vacuum expectation value $(0,0,v)$ breaks $T^1,T^2$ but preserves both $T^3$ and the independent weak hypercharge generator. With canonical kinetic terms its gauge-boson mass matrix in $(W^1,W^2,W^3,B)$ is $g^2{v}^2\mathrm{diag}(1,1,0,0)$. The two massless neutral gauge bosons prevent this field alone from reproducing Standard Model electroweak symmetry breaking.