A new realization of F4 in terms of SO (3) tensor operators (boson realization) is given. For the first time the internal representation (or bosons) does not correspond to the lowest-dimensional representation of the algebra. Also for the exceptional Lie algebra E7 a new realization in terms of SO (3) tensors is presented, this time constructed out of the lowest-dimensional E7 representation. It is shown that those two particular realizations provide a basis in which certain structural zeros of Racah's 6j symbol can be explained.
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