Ricci scalar in terms of vierbein and spin connection I have been trying to derive the following form for the Ricci scalar in terms of vierbein and spin connection
$$R=(e^{\mu a}e^{\nu b}-e^{\mu b}e^{\nu a})(\partial_\mu \omega_{\nu ab}+\omega_{\mu a}^{\hphantom0\hphantom0c} \omega_{\nu cb})$$
With direct substitution, I'm getting a complicated answer that I'm not able to simplify to this. Is there a more direct way to reach this result?
 A: I got it by grinding out the algebra
\begin{equation}
\Gamma^\nu_{\hphantom 0\mu\lambda}=e^\nu_{\hphantom 0 a}\partial_\mu e_\lambda^{\hphantom 0 a}+e^\nu_{\hphantom 0 a}e_\lambda^{\hphantom 0 b}\omega_{\mu\hphantom 0 b}^{\hphantom 0 a}
\end{equation}
\begin{equation}
R^\rho_{\hphantom 0\mu\sigma\nu}=\partial_\sigma\Gamma^\rho_{\hphantom 0\nu\mu}-\partial_\nu\Gamma^\rho_{\hphantom 0\sigma\mu}+\Gamma^\rho_{\hphantom 0\sigma\lambda}\Gamma^\lambda_{\hphantom 0\nu\mu}-\Gamma^\rho_{\hphantom 0\nu\lambda}\Gamma^\lambda_{\hphantom 0\sigma\mu}
\end{equation}
\begin{align}
R^\rho_{\hphantom 0\mu\rho\nu}&\nonumber=\partial_\rho\Gamma^\rho_{\hphantom 0\nu\mu}-\partial_\nu\Gamma^\rho_{\hphantom 0\rho\mu}+\Gamma^\rho_{\hphantom 0\rho\lambda}\Gamma^\lambda_{\hphantom 0\nu\mu}-\Gamma^\rho_{\hphantom 0\nu\lambda}\Gamma^\lambda_{\hphantom 0\rho\mu}\\
&=\partial_\rho e^\rho_{\hphantom 0 a}\partial_\nu e_\mu^{\hphantom 0 a}+e^\rho_{\hphantom 0 a}\partial_\rho\partial_\nu e_\mu^{\hphantom 0 a}+\partial_\rho e^\rho_{\hphantom 0 a}e_\mu^{\hphantom 0 b}\omega_{\nu\hphantom 0 b}^{\hphantom 0 a}+e^\rho_{\hphantom 0 a}\partial_\rho e_\mu^{\hphantom 0 b}\omega_{\nu\hphantom 0 b}^{\hphantom 0 a}+e^\rho_{\hphantom 0 a}e_\mu^{\hphantom 0 b}\partial_\rho\omega_{\nu\hphantom 0 b}^{\hphantom 0 a}\nonumber \\ \nonumber
&-\partial_\nu e^\rho_{\hphantom 0 a}\partial_\rho e_\mu^{\hphantom 0 a}-e^\rho_{\hphantom 0 a}\partial_\nu\partial_\rho e_\mu^{\hphantom 0 a}-\partial_\nu e^\rho_{\hphantom 0 a}e_\mu^{\hphantom 0 b}\omega_{\rho\hphantom 0 b}^{\hphantom 0 a}-e^\rho_{\hphantom 0 a}\partial_\nu e_\mu^{\hphantom 0 b}\omega_{\rho\hphantom 0 b}^{\hphantom 0 a}-e^\rho_{\hphantom 0 a}e_\mu^{\hphantom 0 b}\partial_\nu\omega_{\rho\hphantom 0 b}^{\hphantom 0 a}\\ \nonumber
&+e^\lambda_{\hphantom 0 a}\partial_\nu e_\mu^{\hphantom 0 a}e^\rho_{\hphantom 0 b}\partial_\rho e_\lambda^{\hphantom 0 b}+e^\lambda_{\hphantom 0 a}\partial_\nu e_\mu^{\hphantom 0 a}e^\rho_{\hphantom 0 b}e_\lambda^{\hphantom 0 c}\omega_{\rho\hphantom 0 c}^{\hphantom 0 b}+e^\lambda_{\hphantom 0 a}e_\mu^{\hphantom 0 b}\omega_{\nu\hphantom 0 b}^{\hphantom 0 a}e^\rho_{\hphantom 0 c}\partial_\rho e_\lambda^{\hphantom 0 c}+e^\lambda_{\hphantom 0 a}e_\mu^{\hphantom 0 b}\omega_{\nu\hphantom 0 b}^{\hphantom 0 a}e^\rho_{\hphantom 0 c}e_\lambda^{\hphantom 0 d}\omega_{\rho\hphantom 0 d}^{\hphantom 0 c}\\
&-e^\rho_{\hphantom 0 a}\partial_\nu e_\lambda^{\hphantom 0 a}e^\lambda_{\hphantom 0 b}\partial_\rho e_\mu^{\hphantom 0 b}+e^\rho_{\hphantom 0 a}\partial_\nu e_\lambda^{\hphantom 0 a}e^\lambda_{\hphantom 0 b}e_\mu^{\hphantom 0 c}\omega_{\rho\hphantom 0 c}^{\hphantom 0 b}-e^\rho_{\hphantom 0 a}e_\lambda^{\hphantom 0 b}\omega_{\nu\hphantom 0 b}^{\hphantom 0 a}e^\lambda_{\hphantom 0 c}\partial_\rho e_\mu^{\hphantom 0 c}-e^\rho_{\hphantom 0 a}e_\lambda^{\hphantom 0 b}\omega_{\nu\hphantom 0 b}^{\hphantom 0 a}e^\lambda_{\hphantom 0 c}e_\mu^{\hphantom 0 d}\omega_{\rho\hphantom 0 d}^{\hphantom 0 c}
\end{align}
\begin{align}
R&=e^\mu_{\hphantom 0 e}e^\nu_{\hphantom 0 f}\eta^{ef}R^\rho_{\hphantom 0\mu\rho\nu}\nonumber \\
&=e^\mu_{\hphantom 0 e}e^\nu_{\hphantom 0 f}\eta^{ef}\big[e^\rho_{\hphantom 0 a}e_\mu^{\hphantom 0 b}\partial_\rho\omega_{\nu\hphantom 0 b}^{\hphantom 0 a}-e^\rho_{\hphantom 0 a}e_\mu^{\hphantom 0 b}\partial_\nu\omega_{\rho\hphantom 0 b}^{\hphantom 0 a}+e^\lambda_{\hphantom 0 a}e_\mu^{\hphantom 0 b}\omega_{\nu\hphantom 0 b}^{\hphantom 0 a}e^\rho_{\hphantom 0 c}e_\lambda^{\hphantom 0 d}\omega_{\rho\hphantom 0 d}^{\hphantom 0 c}-e^\rho_{\hphantom 0 a}e_\lambda^{\hphantom 0 b}\omega_{\nu\hphantom 0 b}^{\hphantom 0 a}e^\lambda_{\hphantom 0 c}e_\mu^{\hphantom 0 d}\omega_{\rho\hphantom 0 d}^{\hphantom 0 c}\big]\nonumber\\
&=(e^{\mu a}e^{\nu b}-e^{\mu b}e^{\nu a})(\partial_\mu \omega_{\nu ab}+\omega_{\mu a}^{\hphantom 0\hphantom 0 c}\omega_{\nu cb})
\end{align}
