<PERSON> scalar in presence of torsion
Many authors deriving the field equations for teleparallel gravity (TPG) move from the <PERSON> tensor for TPG:
$R^\rho{}{\mu\lambda\nu} = - \nabla\lambda K^\rho{}{\mu\nu} + \nabla\nu K^\rho{}{\mu\lambda} - K^\rho{}{\sigma\lambda} K^\sigma{}{\mu\nu} + K^\rho{}{\sigma\nu}K^{\sigma}{}{\mu\lambda}$ where $K^\lambda{}{\mu\nu} = \frac{1}{2}(T_\mu{}^\lambda{}\nu + T{\nu}{}^{\lambda}{}\mu - T^\lambda{}{\mu\nu}) $
To the <PERSON> tensor:
$R_{\mu\nu} = - \nabla_\rho K^\rho{}{\mu\nu} + \nabla\nu K^\rho{}{\mu\rho} - K^\rho{}{\sigma\rho} K^\sigma{}{\mu\nu} + K^\rho{}{\sigma\nu}K^{\sigma}{}_{\mu\rho}$
To the <PERSON> scalar:
$R = -T+2\nabla^\mu T_\mu$ where $T = \frac{1}{2}T^\sigma{}{\mu\nu}T{\sigma}{}^{\mu\nu} + \frac{1}{4} T^\sigma{}{\mu\nu}T^{\nu\mu}{}\sigma - T_{\sigma_\mu}{}^\sigma T^{\nu\mu}{}_{\nu} $
(see, e.g., <PERSON>, <PERSON>, and <PERSON>, Eq. 16 or <PERSON>, <PERSON> and <PERSON>, Eq. 15).
I understand the contraction and move from the <PERSON> tensor to the <PERSON> tensor but am having trouble deriving the <PERSON> scalar. Here's what I have so far (and I'd appreciate any help!):
\begin{equation} \begin{aligned} R = R_\mu{}^\mu & = - g^{\mu\nu}\nabla_\rho (T_\mu{}^\rho{}{\nu} + T\nu{}^\rho{}{\mu} - T^\rho{}{\mu\nu})+ g^{\mu\nu}\nabla_\nu(T_\mu{}^\rho{}\rho + T\rho{}^\rho{}\mu -T^\rho{}{\mu\rho} ) - g^{\mu\nu} K^\rho{}{\sigma\rho} K^\sigma{}{\mu\nu} + g^{\mu\nu} K^\rho{}{\sigma\nu}K^{\sigma}{}{\mu\rho} \& = - \nabla_\rho (T_\mu{}^\rho{}^{\mu} + T^{\mu\rho}{}{\mu} - T^\rho{}{\mu}{}^{\mu})+ \nabla^\mu(T_\mu{}^\rho{}\rho + T\rho{}^\rho{}\mu -T^\rho{}{\mu\rho} ) - K ^\rho{}{\sigma\rho} K^\sigma{}{\mu}{}^{\mu} + K^\rho{}{\sigma}{}^{\mu}K^{\sigma}{}{\mu\rho} \end{aligned} \end{equation}
where we've used the definition of contortion and metric compatibility to pull the metric inside the derivative operator. 818