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Is it possible to simplify an expression in vector form, which involves crossproduct and dot product?

I often need to simplify expressions involving cross product and dot product, for example: f = Dot[Cross[Cross[p1 - p, e1], Cross[p2 - p, e2]], Cross[p3 - p, e3]] ...
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Expressing the output product using dot

I have an expression of the form Hbar.H where H and Hbar are vectors. How can I get the ...