Recall that for 3 vectors [tex]a,b,c[/tex], all in [tex]\mathbb R^3[/tex], the vector triple product
[tex]a\times(b\times c)=(a\cdot c)b-(a\cdot b)c[/tex]
So
[tex](v\times w)\times(w\times u)=((v\times w)\cdot u)w-((v\times w)\cdot w)u[/tex]
Also recall the scalar triple product,
[tex]a\cdot(b\times c)[/tex]
which gives the signed volume of the parallelipiped generated by the three vectors [tex]a,b,c[/tex]. When either [tex]a=b[/tex] or [tex]a=c[/tex], the parallelipepid is degenerate and has 0 volume, so
[tex](v\times w)\cdot w=0[/tex]
and the above reduces to
[tex](v\times w)\times(w\times u)=((v\times w)\cdot u)w[/tex]
so that
[tex](u\times v)\cdot[(v\times w)\times(w\times u)]=(u\times v)\cdot((v\times w)\cdot u)w[/tex]
The scalar triple product has the following property:
[tex]a\cdot(b\times c)=b\cdot(c\times a)=c\cdot(a\times b)[/tex]
Since [tex](v\times w)\cdot u[/tex] is a scalar, we can factor it out to get
[tex]((v\times w)\cdot u)((u\times v)\cdot w)[/tex]
and by the property above we have
[tex](u\times v)\cdot w=u\cdot(v\times w)[/tex]
and so we end up with
[tex][u\cdot(v\times w)]^2[/tex]
as required.
