Picture below is from Topping's Lectures on the Ricci flow. I can't understand the (2.1.2).
What I know: for any vector field $Z\in \mathfrak X(M)$, there is $$ R(X,Y)Z = \nabla _Y\nabla _X Z - \nabla _X \nabla _Y Z + \nabla _{[X,Y]}Z $$ Similar, if $\omega\in \mathfrak X(M)^*$ which is dual space of $\mathfrak X(M)$, there is $$ R(X,Y)\omega = \nabla _Y\nabla _X \omega - \nabla _X \nabla _Y \omega + \nabla _{[X,Y]}\omega $$ Therefore, assume $A=A_{i_1,...,i_k} dx^{i_1}\otimes ...\otimes dx^{i_k} $, there is $$ \nabla^2_{Y,X} A(W,Z,...) = (\nabla_Y\nabla _X A)(W,Z,...) - (\nabla_{\nabla_X ~Y} A )(W,Z,...) \\ =\nabla_Y[X(A(W,Z,...))] - \nabla_YA(\nabla_XW,Z,...)-\nabla _Y A(W, \nabla_XZ,...) -... \\ ~~~~~~ -(\nabla_XY)(A(W,Z,...)) +A(\nabla_{\nabla_X~Y}W,Z,...) +A(W, \nabla_{\nabla_X~Y}Z,...) +... $$ similar, I know $-\nabla^2_{X,Y} A(W,Z,...)$. But I don't know how to get the second, third, forth lines of (2.1.2).
