Quaternion
= Quaternion
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A quaternion is $a+bi+cj+dk$ with real coefficients, where $i^2=j^2=k^2=ijk=-1$. They form a noncommutative real <division algebra>. The norm squared is $a^2+b^2+c^2+d^2$, and every nonzero quaternion has inverse its conjugate divided by that squared norm. The unit quaternions form the <Lie group> $S^3$.