© 1969 by London Mathematical Society
Correction to "Two Theorems on Whitehead Products"
University of Liverpool
Abstract
The general Jacobi (or Witt) identity on page 510 of [1] is written down incorrectly.
The correct version is
[[–b,a],c]b + 2[[–c, b], a]c + 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 x B x 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 x C) x B), X],
the element [[a, b], [–c, b]c is zero in [S(A x B x 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] + 2[[b, c], a] + 1[[c, a], b] = 0.
The result stated in Theorem 1 now follows since [S(A x B x 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 x ( B x C)), X]