The action of a massless scalar field in curved spacetime is given by:
\begin{equation} S(\phi)=\int d^{4}x \sqrt{-g}\left(g^{\mu\nu}\phi_{,\mu}\phi_{,\nu}\right) \end{equation}
Now the action can be rewritten using the tetrad formalism as:
\begin{equation} S(\phi)=\int d^{4}x e\left(\eta^{ab}\phi_{,a}\phi_{,b}\right) \end{equation}
where $e$ is the determinant of the veilbein $e^{a}_{\mu}$ and we have used the identity $\phi_{,a}=e^{\mu}_{a}\phi_{,\mu}$.
Is it correct to assume that the equation of motion can be given by:
$$\partial_{a}\left(\frac{\partial\mathcal{L}}{\partial\left(\partial_{a}\phi\right)}\right) - \frac{\partial\mathcal{L}}{\partial\phi} = 0 $$ where $\mathcal{L}$ is the Lagrangian density?
In which case we have explicitly:
$$e \eta^{ab}\phi_{,ab}+\eta^{ab}\phi_{,b}e_{,a}=0$$