We devise an efficient algorithm for the symbolic calculation of irreducible angular momentum and spin (LS) eigenspaces within the n-fold antisymmetrized tensor product. Lambda V-n(u), where n is the number of electrons and u - s, p, d, ... denotes the atomic subshell. This is an essential step for dimension reduction in configuration-interaction (CI) methods applied to atomic many-electron quantum systems. The algorithm relies on the observation that each L-z eigenstate with maximal eigenvalue is also an L-2 eigenstate (equivalently for S-z and S-2), as well as the traversal of LS eigen-states using the lowering operators L- and S-. Iterative application to the remaining states in Lambda V-n(u) leads to an implicit simultaneous diagonalization. A detailed complexity analysis for fixed n and increasing subshell number u yields run time O(u(3n-2)). A symbolic computer algebra implementation is available online.