= SO(4) vector-Higgs gauge-boson spectrum
{c}
{title2=$SO(4)\to SO(3),\quad m_+=0,\quad m_-=|gv|$}
A real four-vector vacuum $\Phi_0=ve_4$ leaves the rotations of its perpendicular three-space unbroken. In a Hermitian generator basis split into two commuting SU(2) triples with $(T^a+T^{a+3})e_4=0$, define $A^{a,\pm}=(A^a\pm A^{a+3})/\sqrt2$. For scalar kinetic term $|D\Phi|^2$ and canonical vector kinetic terms, its vacuum value is $g^2v^2\sum_a(A^{a,-})^2/2$. Comparison with $m^2A^2/2$ gives three massless and three mass-$|gv|$ <gauge bosons>.
The factor multiplying the scalar kinetic term matters: replacing it by $|D\Phi|^2/2$ at fixed numerical $v$ changes the massive value to $|gv|/\sqrt2$. The unbroken global subgroup is SO(3), rather than its simply connected SU(2) cover. The <Spin(4) double cover> accounts for that central identification.
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