Understanding the way of calculating the Nijenhuis tensor on $S^6$

Question

In the example I attached from "Werner Ballmann Lectures on Kähler Manifolds"

I want to understand how he calculated the Nijenhuis tensor at $v,w\in T_pS^6$ to be $N(v, w) = 2\{(p · v) · w − (p · w) · v − p · (v · w) + p · (w · v)\} = 4[p, v, w]$. Isn't for example $$[J_pv,J_pw]=[(p·v)·(p·w)-(p·w)·(p·v)=2(p·v)·(p·w)=-2((p·v)·p)·w=2(p·(v·p))·w=-2(p·(p·v))·w=2((p·p)·v)·w=2v·w ?$$