$X, Y, N$ are vector fields on a riemannian manifold $M$. $\langle X, N \rangle(p)=0 $, $\langle Y, N \rangle(p)=0$ at some point $p$. Show $\langle [X,Y], N \rangle(p)=0$.
I want to use $[X,Y]=XY-YX$, but I don't know how to deal with the inner product.
Please someone check me, I haven't thought about Lie brackets since I barely understood them 10 years ago, but I don't think this is true, even in $\mathbb{R}^3$
Let $X=\partial_y + x\partial_z$ and $Y=\partial_x -y\partial_z$ and $N=\partial_z$. At $p=0$, we have $\langle X,N\rangle (p)=0$ and $\langle Y,N\rangle (p)=0$. But $[X,Y]=-2\partial_z$.