Given two square 2-arrays \(\mathbf{A}\) and \(\mathbf{B}\) with the same size, the anticommutator \(\left\lbrace \mathbf{A}, \mathbf{B} \right\rbrace\) is defined as

and the commutator \(\left[ \mathbf{A}, \mathbf{B} \right]\) is defined as

Since \(\left\lbrace \mathbf{A}, \mathbf{B} \right\rbrace = \left\lbrace \mathbf{B}, \mathbf{A} \right\rbrace\), the anticommutator can also be called the symmetric 2-bracket. Similarly, since \(\left[ \mathbf{A}, \mathbf{B} \right] = -\left[ \mathbf{B}, \mathbf{A} \right]\), the commutator can also be called the alternating 2-bracket.

The symmetric bracket for square 2-arrays can be viewed as the sum of both possible permutations of the inputs for the 2-product. Similarly, the alternating bracket can be viewed as the sum of both possible permutations of the inputs for the 2-product weighted by the sign of the permutation. The triangle product for 3-arrays takes three inputs, so there are six permutations. Given three cubic 3-arrays with the same size, the symmetric 3-bracket \(\left\lbrace \mathbf{A}, \mathbf{B}, \mathbf{C} \right\rbrace\) is defined as

and the alternating 3-bracket \(\left[ \mathbf{A}, \mathbf{B}, \mathbf{C} \right]\) is defined as

Note that

and

Thus, unlike the alternating 2-bracket, the alternating 3-bracket is cyclic.