quantum field theory - Why is only the third component of weak isospin used as a conserved quantity?

Using Noether's theorem

$$\begin{equation} \partial_0 \int d^3x \left(\frac{\partial L}{\partial(\partial_0\Psi)} \delta \Psi \right) = 0 \end{equation}$$

we get three conserved quantites $Q_i$ from global $SU(2)$ symmetry, because the Lagrangian is invariant under infinitesimal transformations of the form $\delta \Psi = i a_i \sigma_i \Psi$. The conserved quantities that follow from the free doublet Lagrangian $L= i\bar{\Psi} \gamma_\mu \partial^\mu \Psi$ are therefore

$$\begin{align} Q_i&= i\bar{\Psi} \gamma_0 \sigma_i \Psi \notag \\ &= \begin{pmatrix} v_e \\ e \end{pmatrix}^\dagger \underbrace{\gamma_0 \gamma_0}_{{=1}} \sigma_i \begin{pmatrix} v_e \\ e \end{pmatrix} \end{align}$$

Why are the conserved quantities that follow from $i=1$ or $i=2$, never mentioned or used? For $i=1$ we have

$$\begin{align} Q_1&= \begin{pmatrix} v_e \\ e \end{pmatrix}^\dagger \sigma_1 \begin{pmatrix} v_e \\ e \end{pmatrix} \notag \\ &= \begin{pmatrix} v_e \\ e \end{pmatrix}^\dagger \begin{pmatrix} 0 & 1 \\1 & 0 \end{pmatrix} \begin{pmatrix} v_e \\ e \end{pmatrix} \notag \\ &= v_e^\dagger e + e^\dagger v_e \end{align}$$

or for $i=3$ we have

$$\begin{align} Q_3&= \begin{pmatrix} v_e \\ e \end{pmatrix}^\dagger \sigma_3 \begin{pmatrix} v_e \\ e \end{pmatrix} \notag \\ &= \begin{pmatrix} v_e \\ e \end{pmatrix}^\dagger \begin{pmatrix} 1 & 0 \\0& -1 \end{pmatrix} \begin{pmatrix} v_e \\ e \end{pmatrix} \notag \\ &= v_e^\dagger v_e - e^\dagger e \end{align}$$

which is the usually used third component of weak isospin.