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j\cdot k=k\cdot j=0  \end{equation}  This is similar to having 3 independant imaginary numbers or basis vectors. If this were an inner product then they are orthoganal but antiparallell with themselves. They form a group under a product with elements ${0,1,i,j,k,-1,-i,-j,-k}$, $ {0,1,i,j,k,-1,-i,-j,-k} $,  this is an Abelian relationship as the non-Abelian properties of the quaternions was removed with the cross interactions.