### Algebraic calculation of the energy eigenvalues for the nondegenerate three-dimensional Kepler-Coulomb potential.

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Attempts to extend our previous work using the octonions to describe fundamental particles lead naturally to the consideration of a particular real, noncompact form of the exceptional Lie group ${E}_{6}$, and of its subgroups. We are therefore led to a description of ${E}_{6}$ in terms of $3\times 3$ octonionic matrices, generalizing previous results in the $2\times 2$ case. Our treatment naturally includes a description of several important subgroups of ${E}_{6}$, notably ${G}_{2}$, ${F}_{4}$, and (the double cover of) $SO(9,1)$. An interpretation of the actions...