I agree that the notation is confusing. Can you possibly show us where you got the definition from? Also you might want to check if its a scalar triple product:
Triple product - Wikipedia, the free encyclopedia
What does "(a×b,c×d,e×f)" mean? axb etc are vectors. What does a triple of vectors mean? What does "(a,b,d)·(c,e,f)-(a,b,c)·(d,e,f)" mean? If "(a, b, c)" is the triple product of vectors as chiro suggests is the " " just multiplication of numbers?
To start off the proof, note:
which come from vector algebra texts.
Then apply to post 1
(axb,cxd,exf) = axb.(cxdxexf)
and continue on as a good excersize. let us know how you make out.