= Solution
<Gauge invariance> of the <scalar potential> implies the infinitesimal identity
$$
\frac{\partial V}{\partial\phi_i}(T^a)_{ij}\phi_j=0.
$$
The irrelevant common factor $i$ from the real representation's infinitesimal rotation has been divided out. Differentiate with respect to $\phi_k$ and evaluate at the <Stationary point>[stationary] <Vacuum state>[vacuum] $v$:
$$
\left.\frac{\partial^2V}{\partial\phi_k\partial\phi_i}\right|_v(T^a)_{ij}v_j
+\left.\frac{\partial V}{\partial\phi_i}\right|_v(T^a)_{ik}=0.
$$
The second term vanishes by <Stationary point>[stationarity], leaving
$$
\boxed{(M_\phi^2)_{ki}(T^av)_i=0.}
$$
Whenever $T^av\ne0$, this is a nonzero vector in the <kernel> of the <Scalar field>[scalar] <Mass matrix>[mass-squared matrix], so it is a zero-eigenvalue direction. It is tangent to the degenerate <Vacuum state>[vacuum] orbit. In a global theory it is a <Goldstone directions in the scalar mass matrix>[Goldstone direction]; in the gauged theory the corresponding would-be <Goldstone boson>[Goldstone field] supplies a <Longitudinal polarization of a massive vector boson>[longitudinal polarization] of a massive vector.
For the specific <Scalar potential>[potential], <Stationary point>[stationarity] at a nonzero <Vacuum state>[vacuum] gives $v^2=m^2/(4\lambda)$, assuming $m^2>0$ and $\lambda>0$. Its <Hessian matrix>[Hessian] is
$$
(M_\phi^2)_{ij}=(-m^2+4\lambda v^2)\delta_{ij}+8\lambda v_iv_j=8\lambda v_iv_j.
$$
The antisymmetry of the <Adjoint representation>[triplet generators] gives $v_i(T^av)_i=0$, verifying the claimed <kernel> directly.
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