Correction to “Two Theorems on Whitehead Products”

The general Jacobi (or Witt) identity on page 510 of [1] is written down incorrectly.† The correct version is τ[[ −b,a],c] b + τ 2 [[−c, b], a] c + τ 3 [[−a, c], b] a = 0. Theorem 1 is correct as stated†, but its proof should read as follows: The commutators [[−b,a],c] b and [[a, b], c] differ in [S(A × B × C), X] by [[a,b],[−c,b]] c ; that is [[−b, a], c] b = [[a, b], c] + [[a, b], [−c, b)] c . Since [a, b] and [−c, b] both lie in the image of the abelian group [S((A × C) × B), X] , the element [[a, b], [−c, b] c is zero in [S(A × B × C), X] . Similarly [[−c, b], a] c = [[b, c], a] and [[−a, c], b] a = [[c, a], b] which proves the identity [[a, b], c] + τ 2 [[b, c], a] + τ 1 [[c, a] , b] = 0. The result stated in Theorem 1 now follows since [S(A × B × C), X] is abelian. Finally I note that a direct proof of Theorem 1 is obtained from the Zassenhaus identity [[a, b], c] + [[c, a], b] + [[b, c], a] = [a, b] + [b, a] c + [c, a] + [b, a] + [b, c] a + [a, c] + [a, b] c + [c, b] a . Any two of the terms on the right‐hand side commute as above, for example [a, c] and [a, b] c both lie in the image of the commutative group [S(A × ( B × C)), X]