= Solution
The three real components form the <Adjoint representation> of $SU(2)$. Around a nonzero vacuum, a local <gauge transformation> rotates the direction of $\phi(x)$ to the third axis. In this <unitary gauge>, the two angular fields are removed from the scalar parametrization and become longitudinal polarizations of two <gauge bosons>, through the <Higgs mechanism>. Thus $\phi=(0,0,v+\eta)^T$ describes the perturbative physical fluctuations, with $v^2=-m^2/\lambda$.
The subgroup generated by $t^3$ fixes $ve_3$, giving \b[$SU(2)\to U(1)$]. Define the physical vector fields
$$
A_\mu=B^3_\mu,\qquad W^\pm_\mu=\frac{B^1_\mu\mp iB^2_\mu}{\sqrt2}.
$$
The <gauge covariant derivative> on the radial configuration is
$$
D_\mu\phi=\begin{pmatrix}-gB^2_\mu(v+\eta)\\gB^1_\mu(v+\eta)\\\partial_\mu\eta\end{pmatrix},
$$
so the scalar <kinetic term> becomes $\frac12(\partial\eta)^2+g^2(v+\eta)^2W^+_\mu W^{-\mu}$.
Here is an exact form of the Lagrangian in these physical fields, up to its additive vacuum constant. Define
$$
f_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu,\qquad
G^\pm_{\mu\nu}=(\partial_\mu\pm igA_\mu)W^\pm_\nu-(\partial_\nu\pm igA_\nu)W^\pm_\mu,
$$
and
$$
H_{\mu\nu}=f_{\mu\nu}+ig(W^+_\mu W^-_\nu-W^-_\mu W^+_\nu).
$$
The minus sign in the printed non-Abelian <gauge field strength> gives precisely these signs. Then
$$
\boxed{\mathcal L=-\frac14H_{\mu\nu}H^{\mu\nu}-\frac12G^+_{\mu\nu}G^{-\mu\nu}
+\frac12\partial_\mu\eta\partial^\mu\eta
+g^2(v+\eta)^2W^+_\mu W^{-\mu}
-\lambda v^2\eta^2-\lambda v\eta^3-\frac\lambda4\eta^4.}
$$
Its quadratic terms give \b[$m_A=0$, $m_{W^+}=m_{W^-}=gv$, and $m_\eta=\sqrt{2\lambda}\,v$]. The two massive vectors have six real polarization degrees of freedom in total; the massless vector has two and the scalar one. Their total of nine equals the original six vector plus three scalar degrees of freedom.
Under the surviving $U(1)$, $W^+$ and $W^-$ have opposite charges and $\eta$ is neutral. The <Yang-Mills theory> supplies $AW^+W^-$ and $AAW^+W^-$ couplings and quartic charged-vector self-interactions. The scalar term supplies $2g^2v\eta W^+W^-$ and $g^2\eta^2W^+W^-$, together with the cubic and quartic scalar self-interactions. There is no direct $\eta AA$ term.
With suitable fermions, this spectrum offers a massless <photon> and massive charged mediators of short-range <weak charged currents>, so it captures part of the weak/electromagnetic pattern. It differs crucially from the <electroweak interaction> of the <Standard Model>: the latter gauges $SU(2)_L\times U(1)_Y$, uses a complex <Higgs doublet>, and breaks to $U(1)_{\rm em}$. Three rather than two generators are broken, leaving a massive neutral <Z boson> as well as $W^\pm$ and a <photon>. Its charge operator $Q=T^3+Y$ and independent hypercharge coupling give a nontrivial <weak mixing angle>. The present theory has no massive neutral mediator, no such mixing angle, and does not reproduce the observed chiral fermion charge assignments or their usual <Yukawa couplings> from the standard doublet/singlet assignments.
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