We devise an efficient algorithm for the symbolic calculation of irreducible angular momentum and spin (LS) eigenspaces within the n-fold antisymmetrized tensor product ^n Vu, 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 Lz eigenstate with maximal eigenvalue is also an L2 eigenstate (equivalently for Sz and S2), as well as the traversal of LS eigenstates using the lowering operators L_ and S_. Iterative application to the remaining states in ^nVu leads to an implicit simultaneous diagonalization. A detailed complexity analysis for fixed n and increasing subshell number u yields run time O(u3n–2). A symbolic computer algebra implementation is available online.