= Solution
Take $D_\mu=\partial_\mu+iK_\mu$, with the <Pauli matrices> acting on the <Higgs doublet>. For <electroweak hypercharge> $Y=1/2$,
$$
K_\mu=\frac g2\tau^aW_\mu^a+\frac{g'}2 B_\mu I_2
=\frac12\begin{pmatrix}gW_\mu^3+g'B_\mu&g(W_\mu^1-iW_\mu^2)\\g(W_\mu^1+iW_\mu^2)&-gW_\mu^3+g'B_\mu\end{pmatrix}.
$$
This specifies every component of the <gauge covariant derivative> for the <electroweak interaction>. Expanding the <gauge-covariant kinetic term> makes its derivative, trilinear and quartic interactions explicit:
$$
(D_\mu\phi)^\dagger D^\mu\phi=(\partial_\mu\phi)^\dagger\partial^\mu\phi
+i\big[(\partial_\mu\phi)^\dagger K^\mu\phi-\phi^\dagger K_\mu\partial^\mu\phi\big]
+\phi^\dagger K_\mu K^\mu\phi,
$$
where the <Pauli matrix multiplication law> gives
$$
\phi^\dagger K_\mu K^\mu\phi=
\frac{g^2}{4}W_\mu^aW^{a\mu}\phi^\dagger\phi
+\frac{g'^2}{4}B_\mu B^\mu\phi^\dagger\phi
+\frac{gg'}2W_\mu^aB^\mu\phi^\dagger\tau^a\phi.
$$
The antisymmetric Pauli contribution vanishes because $W_\mu^aW^{b\mu}$ is symmetric in $a,b$. Reversing the sign convention for $D_\mu$ reverses the linear gauge interactions consistently, without changing the masses.
A nonzero <vacuum expectation value> requires $\mu^2<0$. Minimizing the <Higgs potential> gives $v^2=-\mu^2/\lambda$. By an <gauge transformation> choose
$$
\langle\phi\rangle=\frac1{\sqrt2}\binom0v,\qquad
\phi(x)=\frac1{\sqrt2}\binom0{v+h(x)}
$$
in <unitary gauge>. The <electroweak doublet gauge-boson mass matrix> follows by inserting the <vacuum expectation value> in the <gauge-covariant kinetic term>:
$$
\mathcal L_{\mathrm{mass}}=\frac{v^2}{8}\left[g^2\big(W_\mu^1W^{1\mu}+W_\mu^2W^{2\mu}\big)+(gW_\mu^3-g'B_\mu)^2\right].
$$
Define the charged <electroweak gauge bosons> and the neutral rotation through the <Weinberg angle> by
$$
W_\mu^\pm=\frac{W_\mu^1\mp iW_\mu^2}{\sqrt2},\qquad
s_W=\frac{g'}{\sqrt{g^2+g'^2}},\quad c_W=\frac g{\sqrt{g^2+g'^2}},
$$
$$
Z_\mu=c_WW_\mu^3-s_WB_\mu,\qquad
A_\mu=s_WW_\mu^3+c_WB_\mu,
$$
with inverse $W_\mu^3=c_WZ_\mu+s_WA_\mu$ and $B_\mu=-s_WZ_\mu+c_WA_\mu$. Then
$$
\mathcal L_{\mathrm{mass}}=m_W^2W_\mu^+W^{-\mu}+\frac12m_Z^2 Z_\mu Z^\mu,
\qquad
\boxed{m_W=\frac{gv}{2},\quad m_Z=\frac{v\sqrt{g^2+g'^2}}2,\quad m_A=0.}
$$
The factors differ because $W^+$ and $W^-$ are conjugate fields whereas $Z$ is real. The massless <photon> corresponds to the unbroken <Lie algebra generator> $Q=T_3+Y$, which annihilates $\langle\phi\rangle$. Thus three of the four real <gauge bosons> acquire mass, with $m_W=m_Zc_W$. The three would-be <Goldstone bosons> provide their longitudinal polarizations; the remaining scalar $h$ is the <Higgs boson>.
Introduce the left-handed <lepton> doublet $L=(\nu_{eL},e_L)^T$, with $Y_L=-1/2$, and the right-handed singlet $e_R$, with $Y_R=-1$. Use the <chiral projectors> $P_L=(1-\gamma^5)/2$ and $P_R=(1+\gamma^5)/2$ in the course's convention. The minimal <Standard Model> has no right-handed neutrino. The <gauge-invariant> fermion terms are
$$
\mathcal L_{\ell}=\bar L i\gamma^\mu\left(\partial_\mu+i\frac g2\tau^aW_\mu^a-i\frac{g'}2B_\mu\right)L
+\bar e_Ri\gamma^\mu(\partial_\mu-ig'B_\mu)e_R
-\left(y_e\bar L\phi e_R+y_e^*\bar e_R\phi^\dagger L\right).
$$
The doublet contraction in the <Yukawa interaction> is a singlet, and its total <hypercharge> is $+1/2+1/2-1=0$. A bare term $\bar e_Le_R$ would fail <electroweak gauge invariance>. The <gauge covariant derivatives> above already give all the requested fermion-gauge couplings. In terms of mass eigenstates, $e=gs_W=g'c_W$ and they become
$$
\mathcal L_{\mathrm{CC}}=-\frac g{\sqrt2}\left(W_\mu^+\bar\nu_{eL}\gamma^\mu e_L+W_\mu^-\bar e_L\gamma^\mu\nu_{eL}\right),\qquad
\mathcal L_{\mathrm{em}}=+eA_\mu\bar e\gamma^\mu e,
$$
$$
\mathcal L_Z=-\frac g{c_W}Z_\mu\left[\frac12\bar\nu_{eL}\gamma^\mu\nu_{eL}
+\left(-\frac12+s_W^2\right)\bar e_L\gamma^\mu e_L+s_W^2\bar e_R\gamma^\mu e_R\right].
$$
Thus the <weak charged current> is chiral and the neutrino has zero <electric charge>. The <gauge-invariant electron Yukawa mass> follows from the <Yukawa interaction> after <electroweak symmetry breaking>:
$$
-\frac{v+h}{\sqrt2}\left(y_e\bar e_Le_R+y_e^*\bar e_Re_L\right).
$$
Rephase $e_R$ to make $y_e$ real and positive. It gives
$$
\boxed{m_e=\frac{|y_e|v}{\sqrt2},\qquad \mathcal L_{e,h}=-m_e\bar ee-\frac{m_e}{v}h\bar ee.}
$$
The electron <Dirac mass> is therefore compatible with the original <gauge symmetry> through the <Higgs mechanism>. The neutrino remains massless in this minimal renormalizable lepton sector.
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