Euler's attack on the lemma was flawed precisely because of the (implicit) assumption regarding the veracity of a third powers product principle in $\mathbf Z[\sqrt{-3}]$.

A close reading of Euler's proof shows that he needs this property for numbers in $\mathbf Z[\sqrt{-3}]$ with odd norm, and for such numbers that property in fact is true. That is, numbers in $\mathbf Z[\sqrt{-3}]$ with odd norm have unique factorization. Proving that makes Euler's proof correct. Although it is simpler to pass instead to the slightly larger ring $\mathbf Z[\zeta_3]$ that is a UFD and work there (and use knowledge of its units).

Euler's attack on the lemma was flawed precisely because of the (implicit) assumption regarding the veracity of a third powers product principle in $\mathbf Z[\sqrt{-3}]$.

A close reading of Euler's proof shows that he needs this property for numbers in $\mathbf Z[\sqrt{-3}]$ with odd norm, and for such numbers that property in fact is true. That is, numbers in $\mathbf Z[\sqrt{-3}]$ with odd norm have unique factorization. Proving that makes Euler's proof correct, but it is simpler instead to work in the slightly larger ring $\mathbf Z[\zeta_3]$ that is a UFD and use knowledge of its units to pass back to $\mathbf Z[\sqrt{-3}]$.