I'm trying to figure out the step from equation (19) to equation (20) in this document when $\mathcal{F} = 0$ y $\mathcal{A} = 0$. In this case, equation (19) reduces to
$$ - e^{\alpha \phi} \hat{\omega}^{z} {}{b} \wedge e^{b} = \beta e^{\beta} \partial{\mu} \phi (d x^{\mu} \wedge d x^{4} )$$
using $\hat{e}^{z} = e^{\beta \phi} d x^{4}$ (equation (8)),
$$ e^{\alpha \phi} \hat{\omega}^{z} {}{b} \wedge e^{b} = - \beta e^{\beta} \partial{\mu} ( d x^{\mu} \wedge e^{- \beta \phi} \hat{e}^{z} ) $$ $$\hat{\omega}^{z} {}{b} \wedge e^{b} = - \beta e^{- \alpha \phi} \partial{\mu} d x^{\mu} $$
However, I don't know how to get the equation (20)
$$ \hat{\omega}^{a z}=-\hat{\omega}^{z a}=-\beta e^{-\alpha \phi} \partial^a \phi \hat{e}^z $$
Also, I can't understand how replacing in (16) results in
$$ \hat{\omega}^{a b}=\omega^{a b}+\alpha e^{-\alpha \phi}\left(\partial^b \phi \hat{e}^a-\partial^a \phi \hat{e}^b\right) $$
Could anyone give me a suggestion or observation on how to obtain these results?
From (8), for $\mathcal{A} = 0$ \begin{equation} \hat{e}^{z} = e^{\beta \phi} dx^{4} \end{equation} Equation (19) is \begin{align} -e^{\alpha \phi}\, \hat{\omega}^{z}_{b} \wedge e^{b} &= \beta e^{\beta \phi}\partial_{\mu}\phi \, dx^{\mu} \wedge dx^{4} \\ -e^{\alpha \phi}\, \hat{\omega}^{z}_{b} \wedge e^{b} &= \beta e^{\beta \phi}\partial_{b}\phi \, dx^{b} \wedge dx^{4} \\ -e^{\alpha \phi}\, \hat{\omega}^{z}_{b} \wedge e^{b} &=\beta \partial_{b} \phi \, dx^{b} \wedge \hat{e}^{z} \\ \hat{\omega}^{z}_{b} \wedge e^{b} &= -\beta e^{-\alpha \phi} \partial_{b} \phi \, dx^{b} \wedge \hat{e}^{z} \\ \hat{\omega}^{z}_{b} \wedge e^{b} &= -\beta e^{-2\alpha \phi} \partial_{b} \phi \, e^{b} \wedge \hat{e}^{z} \\ \hat{\omega}^{z}_{b} \wedge e^{b} &= \beta e^{-2\alpha \phi} \partial_{b} \phi \, \hat{e}^{z} \wedge e^{b} \\ \end{align} This means that \begin{align} &\hat{\omega}^{z}_{b} = \beta e^{-2\alpha \phi} \partial_{b} \phi \,\hat{e}^{z} \\ &\hat{\omega}^{zb} = \beta e^{-\alpha \phi} \partial^{b} \phi \,\hat{e}^{z} \\ &\boxed{\hat{\omega}^{bz} = -\beta e^{-\alpha \phi} \partial^{b} \phi \,\hat{e}^{z}} \end{align}