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

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

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

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

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

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

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

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