= Solution
Write $H=2^{-1/2}e^{-i\xi^aT^a}(0,v+h)^{\mathsf T}$. Differentiating the exponential to first order in the fields, or using its Maurer-Cartan form exactly, shows that the angular fields enter $D_\mu H$ through
$$
\partial_\mu\xi^aT^a+gW_\mu^aT^a+\frac{g'}2B_\mu
$$
with higher-order commutator terms dictated by the same group-valued combination. A local $SU(2)_L$ transformation can set $\xi^a=0$. In this <unitary gauge>,
$$
H=\frac1{\sqrt2}\begin{pmatrix}0\\v+h\end{pmatrix}.
$$
The three Goldstone modes have become the longitudinal polarizations of the three massive electroweak gauge bosons, which is the <Higgs mechanism>.
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