= Solution
Assume $\lambda>0$ and $v>0$. The <vacuum manifold> is the sphere $|\phi|=v$. Choose $\phi_0=ve_3$. For the given generators, $t^3\phi_0=0$ whereas $t^1\phi_0$ and $t^2\phi_0$ are nonzero and independent. The <Adjoint representation> of <SU(2)> rotates this sphere, and the full connected <stabilizer subgroup> of $ve_3$ is generated by $t^3$. Hence the classical symmetry pattern is
$$
\boxed{SU(2)\longrightarrow U(1).}
$$
The central element of <SU(2)> acts trivially on the triplet and belongs to this stabilizer; it does not add a separate broken direction. The gauge-field-free representative has zero <gauge field strength> and constant scalar magnitude. Locally near this nonzero <classical vacuum>, <unitary gauge> aligns the triplet along the third axis.
First make the gauge normalization explicit. Write the gauge <kinetic term> as $-\mathcal N F^a_{\mu\nu}F^{a\mu\nu}/4$, where $\operatorname{Tr}(t^at^b)=\mathcal N\delta^{ab}$ if $F=F^at^a$ is the matrix used inside the printed trace. <Canonical normalization of a gauge kinetic term> gives
$$
\mathcal A^a_\mu=\sqrt{\mathcal N}\,A^a_\mu,\qquad g_c=\frac{g}{\sqrt{\mathcal N}},\qquad
\mathcal F^a_{\mu\nu}=\partial_\mu\mathcal A^a_\nu-\partial_\nu\mathcal A^a_\mu-g_c\epsilon^{abc}\mathcal A^b_\mu\mathcal A^c_\nu.
$$
For a conventionally normalized component trace, $\mathcal N=1$ and $g_c=g$. If the trace is the ordinary matrix trace in the displayed three-dimensional representation, $\operatorname{Tr}(t^at^b)=2\delta^{ab}$ and $\mathcal N=2$. The PDF does not specify which trace convention is intended; both are covered by this formula.
In <unitary gauge>, the <gauge covariant derivative> is
$$
D_\mu\phi=\bigl(-g_c(v+\eta)\mathcal A^2_\mu,\quad g_c(v+\eta)\mathcal A^1_\mu,\quad\partial_\mu\eta\bigr)^T.
$$
Expanding this <kinetic term> and the <scalar potential> gives the complete physical-field <Lagrangian>
$$
\mathcal L=-\frac14\mathcal F^a_{\mu\nu}\mathcal F^{a\mu\nu}
+\frac12(\partial_\mu\eta)(\partial^\mu\eta)
+\frac{g_c^2}{2}(v+\eta)^2\bigl[(\mathcal A^1_\mu)^2+(\mathcal A^2_\mu)^2\bigr]
-\frac{\lambda v^2}{2}\eta^2-\frac{\lambda v}{2}\eta^3-\frac\lambda8\eta^4.
$$
Here a square of a vector means its contraction with the <Minkowski metric>, in signature $(+---)$. The labels 1 and 2 represent massive vectors; 3 represents the surviving Abelian vector.
To display their charge and all interactions more clearly, put $a_\mu=\mathcal A^3_\mu$, $W^\pm_\mu=(\mathcal A^1_\mu\mp i\mathcal A^2_\mu)/\sqrt2$, and define
$$
f_{\mu\nu}=\partial_\mu a_\nu-\partial_\nu a_\mu,\qquad
C_{\mu\nu}=W^+_\mu W^-_\nu-W^+_\nu W^-_\mu,
$$
$$
G^\pm_{\mu\nu}=(\partial_\mu\pm ig_ca_\mu)W^\pm_\nu-(\partial_\nu\pm ig_ca_\nu)W^\pm_\mu.
$$
With the sign of <gauge field strength> printed in the paper, $\mathcal F^3=f+ig_cC$ and $G^\pm=(\mathcal F^1\mp i\mathcal F^2)/\sqrt2$. Thus the same <Lagrangian> becomes
$$
\boxed{\begin{aligned}
\mathcal L={}&-\frac14(f_{\mu\nu}+ig_cC_{\mu\nu})(f^{\mu\nu}+ig_cC^{\mu\nu})
-\frac12G^+_{\mu\nu}G^{-\mu\nu}
+\frac12(\partial\eta)^2+g_c^2(v+\eta)^2W^+_\mu W^{-\mu}\\
&-\frac{\lambda v^2}{2}\eta^2-\frac{\lambda v}{2}\eta^3-\frac\lambda8\eta^4.
\end{aligned}}
$$
The \b[physical masses] are
$$
\boxed{m_a=0,\qquad m_{W^+}=m_{W^-}=g_cv,\qquad m_\eta=\sqrt\lambda\,v.}
$$
In particular, literal adjoint matrix trace gives $m_W=gv/\sqrt2$; the standard canonical component convention gives $m_W=gv$. These are descriptions with differently normalized couplings, not different physical spectra.
The complex vector pair carries opposite charges under the unbroken <U(1) gauge symmetry>. The <gauge field strength> terms contain $aW^+W^-$ cubic interactions, $aaW^+W^-$ quartic interactions, and four-vector interactions involving the charged fields. There is no pure Abelian cubic or quartic self-interaction. The <Higgs mode> $\eta$ has cubic and quartic scalar interactions and couples through $2g_c^2v\eta W^+W^-+g_c^2\eta^2W^+W^-$. It is neutral and has no tree-level $\eta aa$ interaction. This is the <physical charged-vector Lagrangian for an adjoint SU2 Higgs model>.
If the angular fields $\pi_1$ and $\pi_2$ were retained, their vanishing potential masses would identify the two <Goldstone bosons> of the ungauged triplet. In the <gauge theory> they mix with the broken-direction <gauge fields> and can be removed by <gauge fixing>; the <Higgs mechanism> uses them as the longitudinal polarizations of the two massive vectors. They are not additional physical massless scalars. The physical degrees of freedom are conserved: $3+3\times2=9$ before rearrangement, and $1+2\times3+2=9$ afterwards.
\b[This is not the <Standard Model> <electroweak interaction>.] It has three original <gauge bosons>, leaving two massive charged vectors and one massless neutral vector, with no massive neutral <Z boson>. The <Standard Model> instead has $SU(2)_L\times U(1)_Y$, a complex <Higgs doublet>, and three massive vectors plus the <photon>. Its charge is $Q=T_3+Y$, not just the surviving $T_3$. Adding <fermions> cannot supply the missing gauge generator or turn the surviving neutral vector into both a <photon> and a <Z boson>. In particular, the usual right-handed <fermions> in the <Standard Model> are <SU(2)> singlets; they would have zero charge if only $T_3$ were available, instead of the charges produced by <hypercharge>.
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