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Ternary algebras, constructed from ternary commutators, or as we call them, ternutators, defined as the alternating sum of products of three operators, have been shown to satisfy cubic identities as necessary conditions for their existence. Here we examine the situation where we permit identities not solely constructed from ternutators or nested ternutators and we find that in general, these impose additional restrictions; for example, the anti-commutators or commutators of the operators must obey some linear relations among themselves. [ABSTRACT FROM AUTHOR]