$R$ denotes the Riemann curvature operator $R(X,Y)Z:=\nabla_{X}(\nabla_{Y}Z)-\nabla_{Y}(\nabla_{X}Z)-\nabla_{[X,Y]}Z$
I am trying to understand why $\langle R(X,Y)Z,W\rangle=-\langle R(X,Y)W,Z\rangle$ is equivalent to saying $R(X,Y)$ is a rotation? My guess is that it somehow implies that $R(X,Y)$ is somehow an orthogonal map.
Regarding the second question in the title there is unfortunately little I can write since there is not much about it in the lecture notes.