In the picture below ,I don't know how to contract the $M_{\alpha\beta}$. I just used to the contract in Lee's book $$ tr:T^{k+1}_{l+1}\rightarrow T^k_l \\ F\rightarrow trF(\omega^1...\omega^l,\cdot,V_1...V_k,\cdot) $$ In fact, I am very strange with $\mathcal{P}(M_{\alpha\beta},h_{\alpha\beta})$. Besides, in the second red line why $N_{\alpha\beta}$ can be made so ?
Picture below is from the 212 page of this.
