I am trying to prove the vector identity $A\times(B\times C)$ for my students in a "geometrical", but elementary, way (A, B and C are non-orthogonal vectors in 3D Cartesian space). B and C define a plane, and I denote the projection of A on that plane by a. I'd like to find an expression for the vector $a^\prime$ that is orthogonal to a in terms of the vectors A, B and C. Introducing Cartesian coordinates in the BC plane, where x is along B and y is orthogonal to x, I am able to show that $a^\prime = [ (A\cdot C)B - (A\cdot B)C ] / |B\times C|$ (as it must because $a^\prime |B\times C|$ is just $A\times(B\times C)$) - but the method is clumsy.
The result for $a^\prime$ is very "cross-producty looking" and I have been staring at it hoping to "see" in a simple way that it must be the case. Having had no luck at that, I now am hoping that someone cleverer than me knows a "coordinate-free" way to just write down the result for $a^\prime$ (i.e. without introducing the auxiliary x-y coordinates). Thanks.