Let $M$ be a surface in $\mathbb{R}^3$ given by a regular chart, say $X:M \longrightarrow \mathbb{R}^3$, with its first fundamental form $g$, Gauss map $N$, Gaussian curvature $K$ and mean curvature $H$. I want to show that $\Delta N = -2g^{ij} \partial_i H \partial_j X - 2(2H^2-K)N,$ where $\Delta$ denotes the Laplace-Beltrami operator on $M$.

Expanding $\Delta N$ in terms of the basis $(\partial_1 X,\partial_2 X, N)$, I have already shown that the $N$ component is in fact $-2(2H^2-K)$, but unfortunately I seem to unable to calculate the rest. I've been looking for the result in several books on differential geometry but have not been successful so far. Can someone give a hint or just a reference for a proof of this equation?