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A new homotopy fibration is constructed at the prime 3 which shows that the quotient group E7/F4 is spherically resolved. This is then used to show that the 3-primary homotopy exponent of E7 is bounded above by 323, which is at most four powers of 3 from being optimal.

An upper bound for the 3-primary homotopy exponent of the exceptional Lie group $E_7$