  In this paper we study ternary algebras of third-order hypermatrices. By
hypermatrix we mean a complex-valued variable with three indices, which is also
called a three-dimensional matrix or spatial matrix. We assume that a
hypermatrix is defined in three-dimensional Euclidean space and when this space
is rotated, it transforms as a SO(3)-tensor. We consider two ternary
multiplications of hypermatrices, which have the property of generalized
associativity. We explore the geometric meaning of two independent
SO(3)-invariants of hypermatrices and show that one of them defines a Hermitian
metric. We study the 10-dimensional subspace of hypermatrices, known in the
theory of representations of the rotation group, as the space of the weight 2
tensor representation of the rotation group. It is proved that the elements of
this subspace that satisfy the regularity condition are right biunits of the
ternary algebra of hypermatrices. The motivation for studying biunits of
ternary algebra of hypermatrices was a ternary generalization of the Pauli
exclusion principle.
