I am reading ch 7 of Geometry, Topology, and Physics by Mikio Nakahara. And I am trying to understand how to prove the second Bianchi identity: $$ \left(\nabla_X R\right)(Y, Z) V+\left(\nabla_Z R\right)(X, Y) V+\left(\nabla_Y R\right)(Z, X) V=0 $$
In his approach, the symmetrizer is defined as $$ \mathfrak{S}\{f(X, Y, Z)\}=f(X, Y, Z)+f(Z, X, Y)+f(Y, Z, X) $$ Since the assumption is torsion-free connection: $$ T(X, Y)=\nabla_X Y-\nabla_Y X-[X, Y]=0 $$ we have $$R(T(X,Y),Z)V = 0$$ and $$\mathfrak{S}(R(T(X,Y),Z))) = 0$$ Equation (7.82) on the book: $$ \begin{aligned} 0 &=\mathfrak{S}\left\{R\left(\nabla_X Y, Z\right)-R\left(\nabla_Y X, Z\right)-R([X, Y], Z)\right\} V \\ &=\mathfrak{S}\left\{R\left(\nabla_Z X, Y\right)+R\left(X, \nabla_Z Y\right)-R([X, Y], Z)\right\} V . \end{aligned} $$
I failed to understand how the first equality lead to the second equality in equation (7.82).
Any suggestions? Thanks in Advance!