I want to calculate the Ricci scalar using the Cartan equations in the context of reducing a spatial dimension of a 4-dimensional theory describing an empty spacetime with a metric of the form
\begin{equation} d \hat{s} = e^{2 \alpha \varphi} d s_{3}^{2} + e^{2 \beta \varphi} d \phi^{2} \end{equation}
where the quantities with hats belong to four-dimensional spacetime and $d \phi$ is the dimension to be compactified. From the metric, I took the Vielbein basis:
\begin{equation} \hat{e}^{a} = e^{\alpha \varphi} e^{a}, \ \ \ \hat{e}^{3} = e^{\beta \varphi} d \phi \end{equation}
For this, I first set the indices $M, N, \cdots$ as 4-dimensional curved indices and $\mu, \nu, \cdots$ as 3-dimensional curved indices. Also, $A, B, C, \cdots$ are 4-dimensional flat indices and $a,b,c, \cdots$ are 3-dimensional flat indices. Then, to calculate the Ricci scalar I employed Cartan structural equations are
\begin{align*} \hat{T}^{A} &= d \hat{e}^{A} + \hat{\omega}^{A} {}_{B} \hat{e}^{B} \\ \hat{R}^{A} {}_{B} &= d \hat{\omega}^{A} {}_{B} + \hat{\omega}^{A} {}_{C} \wedge \hat{\omega}^{C} {}_{B} \end{align*}
where $\hat{T}^{A}$ is the torsion and $\hat{R}^{A} {}_{B}$, the curvature. I have already calculated the non-zero components of the spin connection and they are given by
\begin{align} \hat{\omega}^{a} {}_{b} &= \omega^{a} {}_{b} + \alpha e^{- \alpha \varphi} \left( \partial_{b} \varphi \hat{e}^{a} - \partial^{a} \varphi \hat{e}_{b} \right) \\ \hat{\omega}^{3} {}_{a} &= \beta e^{-\alpha \varphi} \partial_{a} \varphi \hat{e}^{3} \end{align}
According to equation (5.17) of this document, the Ricci scalar is obtained from $\hat{R} = \eta^{AB} \hat{R}_{AB} = \eta^{ab} \hat{R}_{ab} + \hat{R}_{33}$. I started by calculating $\hat{R}_{33}$ with the curvature of the Cartan equations,
\begin{align} \hat{R}_{33} &= d \hat{\omega}_{33} + \hat{\omega}^{3}_{a} \wedge \hat{\omega}^{a}_{3} = \hat{\omega}^{3}_{a} \wedge \hat{\omega}^{a}_{3} \\ &= \left( \beta e^{- \alpha \varphi} \partial^{a} \varphi \hat{e}_{3} \right) \wedge \left( \beta e^{- \alpha \varphi} \partial_{a} \varphi \hat{e}^{3} \right) = \beta^{2} e^{- 2 \alpha \varphi} \left( \partial^{a} \varphi \hat{e}_{3} \wedge \partial_{a} \varphi \hat{e}^{3} \right) \end{align}
where I used the fact $\hat{\omega}_{33} = 0$ and replaced the second component of the spin connection. However, what I should get is
\begin{equation} \hat{R}_{33} = \beta^{2} e^{- 2 \alpha \varphi} \square \varphi \end{equation}
where $\square \varphi = \partial_a \partial^{a} \varphi = \partial_{\mu} \partial^{\mu} \varphi$ (according to equation 23 in this document where for us $\mathcal{F} = 0$ and $z = 3$). Where is my mistake or what did I miss? On the other hand, I tried to calculate the $\hat{R}_{ab}$ component also using the curvature in the Cartan equations,
\begin{equation} \hat{R}^{a}_{b} = d \hat{\omega}^{a} {}_{b} + \hat{\omega}^{a} {}_{c} \wedge \hat{\omega}^{c} {}_{b} + \hat{\omega}^{a} {}_{3} \wedge \hat{\omega}^{3} {}_{b} \end{equation}
where
\begin{align} d \hat{\omega}^{a} {}_{b} &= d \hat{\omega}^{a} {}_{b} + d \left[ \alpha e^{- \alpha \varphi} \left( \partial_{b} \varphi \hat{e}^{a} - \partial^{a} \varphi \hat{e}_{b} \right) \right] \\ \hat{\omega}^{a} {}_{c} \wedge \hat{\omega}^{c} {}_{b} &= \left[ \omega^{a} {}_{c} + \alpha e^{- \alpha \varphi} \left( \partial_{c} \varphi \hat{e}^{a} - \partial^{a} \varphi \hat{e}_{c} \right) \right] \wedge \left[ \omega^{c} {}_{b} + \alpha e^{- \alpha \varphi} \left( \partial_{b} \varphi \hat{e}^{c} - \partial^{c} \varphi \hat{e}_{b} \right) \right] \\ \hat{\omega}^{a} {}_{3} \wedge \hat{\omega}^{3} {}_{b} &= \left( - \beta e^{- \alpha \varphi} \partial^{a} \varphi \hat{e}_{3} \right) \wedge \left( \beta e^{- \alpha \varphi} \partial_{b} \varphi \hat{e}^{3} \right) \end{align}
However, when I put everything together I don't get the correct expression for $d \hat{\omega}^{a} {}_{b}$. According to this document (equation 25), what I should get is
\begin{equation} \hat{R}^{a} {}_{b} = e^{- 2 \alpha \varphi} \left( R_{ab} + \left( \alpha \beta - 2 \alpha^{2} \right) \partial_{a} \varphi \partial_{b} \varphi - \alpha \eta_{ab} \square \varphi \right) \end{equation}
Could someone please guide me or make some observations? I would appreciate it if you could correct me or suggest more material because I still haven't gotten used to this type of notation and I can't make any progress.
