In Paul Langacker's The Standard Model and Beyond, equation 3.50 states that, for a Lagrangian $\mathcal{L} = \mathcal{L}_0 + \mathcal{L}_1$, where $\left[ T^i , \mathcal{L}_0 \right] = 0$ and $\left[ T^i , \mathcal{L}_1 \right] \ne 0$, we would have
$$ \mathcal{L}' - \mathcal{L} = \left[ - \mathrm{i} \beta ^i T^i , \mathcal{L}_1 \right] = - \beta ^i \partial ^{\mu} J_{\mu}^i $$
For $J_{\mu}^i$ defined as following
$$ J_{\mu}^i = - \mathrm{i} \frac{\delta \mathcal{L}}{\delta \partial^{\mu} \Phi _a} L_{a b}^i \Phi _b - \mathrm{i} \frac{\delta \mathcal{L}}{\delta \partial^{\mu} \Phi _a^{\dagger}} \left( - L_{a b}^{i *} \right) \Phi _b^{\dagger} $$
I completely have no clue on how to prove the second equality in the first equation, could anyone help me on that?
