Can somemone help me prove the following identity? $$ \| \nabla_X(J)(Y) \|^2 = \langle R_{XY}X,Y\rangle - \langle R_{XY}JX,JY\rangle$$ where $J$ is the almost complex structure, and $R$ the curvature tensor $$ R_{XY} = \nabla_{[X,Y]} - [\nabla_X,\nabla_Y]. $$ It is from Gray's paper "Almost complex submanifolds of the six sphere" which is available here: http://www.ams.org/journals/proc/1969-020-01/S0002-9939-1969-0246332-4/S0002-9939-1969-0246332-4.pdf
The paper gives some hints as how to prove this, but I am stuck in the middle and would like to know if there is a simpler way to do it.
Also, general hints on how to prove identities of this kind are appreciated.
Edit: A nearly Kähler manifold is an almost Hermitian manifold satisfying $$ \nabla_X(J)(X)=0 $$ for all vector fields $X$. An almost Hermitian manifold is an almost complex manifold satisfying $$ \langle JX,JY \rangle = \langle X, Y\rangle $$ for all vector fields $X,Y$.