A lie-algebraic formulation for triply orthogonal and general coordinate systems in three-dimensional euclidean and lorentz spaces

We give a Lie-algebraic formulation for the interacting geometries of orthogonal families of coordinate surfaces in 3-dimensional Euclidean- And Lorentz-orthogonal coordinate systems. A study of the Gauss-Lamé equations and their variational equations in this setting leads to formulas for constructing more general 3-dimensional coordinate transformations. To motivate the general constructions, we begin with special cases of orthogonal coordinate systems in 3-dimensional Lorentz space, built from orthogonal systems in the plane.