Consider a closed, compact, embedded surface $f:M \rightarrow \mathbb{R}^3$ and a vectorfield $X$ on the surface that can be decomposed in the surface frame basis $\{e_1,e_2,e_3\}$, where $\{e_1,e_2\}$ is basis of the tangent bundle and $e_3 = N$ is the surface normal field. Denote $X = X_T + X_N$ with surface tangent part $X_T$ and surface normal part $X_N$.
Now, if we consider the (vector) Laplace Beltrami operator $\Delta$ acting on X (i.e. component wise 1D Laplace Beltrami operator), what can be said about the tangent/normal decomposition of $\Delta X_T$ and $\Delta X_N$?
Can one show what this decomposition looks like using the concise exterior calculus notation $\Delta := *d*d$?