Exercise 7, § 3

Suppose given on a set E an (associative and commutative) addition under which all the elements of E are invertible and an (associative) multiplication which is doubly distributive with respect to addition (in other words, a ring structure); write $[x, y] = xy - yx$; the law $(x, y) \mapsto [x, y]$ is doubly distributive with respect to addition. For x and y to be permutable under multiplication, it is necessary and sufficient that $[x, y] = 0$; and we have the identities

(the second is known as the "Jacobi identity"). The second identity may also be written

which expresses the "deviation from associativity" of the law $[x, y]$.