A cross product identity?

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In $\Bbb R^3$ if $A \times B = C$ and $B \times C = A$.

In what circunstances we can say that $A \times C = B$ ? Is it the only possibility? Why (or why not?)

Intuitively (geometrically) I think that the equation holds, but I also think that this is not true, rigorously speaking.