  Using the embedded gradient vector field method (see P. Birtea, D. Comanescu,
Hessian operators on constraint manifolds, J. Nonlinear Science 25, 2015), we
present a general formula for the Laplace-Beltrami operator defined on a
constraint manifold, written in the ambient coordinates. Regarding the
orthogonal group as a constraint submanifold of the Euclidean space of $n\times
n$ matrices, we give an explicit formula for the Laplace-Beltrami operator on
the orthogonal group using the ambient Euclidean coordinates. We apply this new
formula for some relevant functions.
