Following a research direction proposed in an earlier work, the ternary octonion algebra \(\mathfrak {O}\) , which is a ternary composition algebra, is considered. By hand and applying computational linear algebra on matrices, 1-identities and 2-identities of \(\mathfrak {O}\) are established. From some of these identities, the non-conservativeness of \(\mathfrak {O}\) and of some of its binary reduced algebras, which are binary standard composition algebras of types II and III, is proved. Also from identities of \(\mathfrak {O}\) , using computational linear algebra based on the representation theory of the symmetric group, ternary enveloping algebras for ternary Maltsev algebras are constructed.
