How is this proof valid? ax(bxc) = (c.a)b-(b.a)c

mr_coffee

Hello everyone, i'm looking at this proof and its making sence until the very last part, and it doens't friggin' work...the terms don't match up... Also is the notation I used correct in the proof, like does there have to be any vector notation I don't have or do you see anyhting you don't like? I'm proving the "The Vector triple product"
Here is my proof:
Proof
Look at the 2nd part of ax(bxc) -----> a1b1c3 - a2b22c3 + a2b3c2
then look at the last part (c.ad)b - (b.a)c ---> a1b3c1 - a1b1c3 + a2b3c2 - a2b2c3

You see only 1 term isn't match up!! the a1b1c3 not equal to a1b3c1 whats going on? Thanks

mr_coffee

Nevermind, my friend found the problem! I forgot a term in ax(bxc) part.