Exactly solvable Schrödinger equations play important roles in theoretical physics. Most of them are known to enjoy some remarkable algebraic structures, such as supersymmetry, shape invariance, or existence of annihilation/creation operators. In this talk I shall present yet another (possibly new) Lie-algebraic approach to exactly solvable bound-state problems in one-dimensional quantum mechanics. I shall revisit five specific examples: Coulomb, Rosen-Morse, Manning-Rosen, Eckart, and Liouville potentials, all of whose bound-state problems are exactly solvable. I shall discuss that these bound-state problems can be solved by making use of unitary representations of the Lie algebras so(2,1), so(3), or iso(2) in a little bit strange way.