Milnor's octahedron

Question

If $V$ is a vector space, then a curvaturelike tensor in $V$ is a quadrilinear map $F:V \times V \times V \times V \to \Bbb R$ satisfying

1) $F(x,y,z,w) = -F(y,x,z,w) = -F(x,y,w,z)$;

2) $F(x,y,z,w) = F(z,w,x,y)$;

3) $F(x,y,z,w) + F(y,z,x,w) + F(z,x,y,w) = 0$,

for all $x,y,z,w \in V$. It turns out that the Bianchi identity (item 3) actually implies item 2, assuming 1. This implication is called "Milnor's octahedron argument".

I'd like to know where this appeared for the first time. Thanks!

Answer 1

I don't know if it's truly the first appearance, but it seems likely the name originates from Milnor's classic 1963 book Morse Theory, in which you can find the following illustrated argument:

Here (1), (2) and (3) are your 1) and 3), with (4) being your conclusion 2).