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$$(a \times b) \times (c \times d)= [a,b,d]c - [a,b,c]d.$$

I understand $[a,b,d] = a \cdot (b \times d)$ and $[a,b,c] = a \cdot (b \times c)$ but not really sure where else to go with this.

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YES! Thank you that works. – Apeman Aug 11 '12 at 22:09

As Rahul mentioned: $$n \times (m \times o) = (n \cdot o)m - (n \cdot m)o$$

Then it follows: $$(a \times b) \times (c \times d) = (a \times b \cdot d)c -(a \times b \cdot c)d$$ Using the expressions you mentioned above this can be further simplified to:$$[a,b,d]c - [a,b,c]d$$

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