Derive and Verify Vector Identities 

Famous three-dimensional identities.

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X TensorExpand[(v\[Cross]w).v == 0]

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X TensorExpand[a\[Cross](b\[Cross]c) == b (a.c) - c (a.b)]

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X TensorExpand[(a\[Cross]b).(c\[Cross]d) == a.c b.d - a.d b.c]

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Less famous three-dimensional identities.

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X TensorExpand[ a (b\[Cross]c).d - b (a\[Cross]c).d + c (a\[Cross]b).d - d (a\[Cross]b).c == 0]

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X TensorExpand[(a\[Cross]b)\[Cross](c\[Cross]d)]

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Two four-dimensional identities.

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X TensorExpand[a\[Cross]b\[Cross]c == b\[Cross]c\[Cross]a]

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X TensorExpand[(a\[Cross]b\[Cross]c).(d\[Cross]e\[Cross]f)]

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