Doran's book Geometric Algebra for Physicist

I am a newbie in GA. I am going through the Doran’s book and I stumped onto the demonstration the outer product is associative in section 2.4

For a start it decomposes

a\wedge(b\wedge c)=\frac 1 2 (\,a(b\wedge c)+(b \wedge c)a\,)

However I dont understand the reason for the + symbol.

According to the formulaes in section 2.1:

p\wedge q=\frac 1 2 (pq-qp)

So calling a=p and b\wedge c=q

the abobe formula should be:

a\wedge(b\wedge c)=\frac 1 2 (\,a(b\wedge c)\textcolor{red}{-}(b \wedge c)a\,)

Problem is whith + the demonstration works, and whith the - does not.

The formula a \wedge b = (ab-ba)/2 only applies if a and b are vectors by the way, it’s not a general formula. The most general form of that expression is more complicated and involves a sum of more terms for a grade 3 element, and the formula simply does not apply to grade 3 and higher products.

You can find the most general form of the expression if you study my notes at Algebra · Grassmann.jl

Doran and a lot of other people fail to tell people that these formulas only apply in limited context, and are not the most general expression.

For a reasonably thorough discussion of these kinds of questions let me recommend Chisolm’s “book” on arxiv, [1205.5935] Geometric Algebra