I have the following exercise:
Let $E=M \times \mathbb{R}^N$, with connection $\nabla:=d+A$, $d(X,s):=ds(X)=X(s)$ (my interpretation),with $A \in \Omega^1(M, gl_N(\mathbb{R}))$. We have to show that
$R^{\nabla}(X,Y)= \underbrace{X(A(Y))-Y(A(X))}_{\text{to be understood component wise}}-A([X,Y])+\underbrace{[A(X),A(Y)]}_{=A(X)A(Y)-A(Y)A(X)}$
So far I have for $s \in \Gamma(E)$:
$(d+A)_X(d+A)_Y(s)=X(Y(s))+X(A(Y)s)+A(X)Y(s)+A(X)A(Y)(s)$
which implies
$R^{\nabla}(X,Y)= \nabla_X\nabla_Y-\nabla_Y\nabla_X-\nabla_{[X,Y]}=X(Y(s))+X(A(Y)s)+A(X)Y(s)+A(X)A(Y)(s)-Y(X(s))-Y(A(X)s)-A(Y)X(s)-A(Y)A(X)(s)-[X,Y](s)-A([X,Y])(s)=[A(X),A(Y)](s)-A([X,Y])s+X(A(Y)s)-Y(A(X)s)+\underbrace{A(X)Y(s)-A(Y)X(s)}_{\text{disturbing term}}$
Now I don't know how to get rid of the last two terms, since without them my proof would be done..
What am I doing wrong? I'm just starting the exercise over and over again and just don't see where my fault is...