How to translate this equation into physicist's notation?

Question

How to translate this equation into physicist's notation, i.e. tensors with indices? $$\left\langle R_{N}\left(u,v\right)w,z\right\rangle =\left\langle R_{M}\left(u,v\right)w,z\right\rangle +\left\langle II\left(u,z\right),II\left(v,w\right)\right\rangle -\left\langle II\left(u,w\right),II\left(v,z\right)\right\rangle $$ This equation is from the Wikipedia article about second fundamental form, last equation of that article. $R_N$ is curvature tensor of a manifold embedded in another manifold with curvature tensor $R_M$. $II$ is the second fundamental form.