Gauss equation with reversed curvature convention ($⟨R^M(X,Y)Z,W⟩=⟨R^V(X,Y)Z,W⟩-⟨II(X,W),II(Y,Z)⟩+⟨II(Y,W),II(X,Z)⟩$)

Here $R(X,Y)=D_{[X,Y]}-[D_X,D_Y]$, the negative of the convention in the ordinary Gauss equation in a curved ambient manifold. Insert the Gauss formula and Weingarten formula into this definition and project tangentially to obtain the displayed identity. Positive sectional curvature uses $⟨R(X,Y)X,Y⟩/|X\wedge Y{|}^2$ with this convention. Thus a Euclidean surface still has Gaussian curvature $K=\det A_{\xi }$; reversing the operator sign without changing the slot order would give the wrong sign.