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Given any three $a,b,c \in \mathbb{H}$ which can be parametrized by some $x \in \mathbb{H}$ as $$\begin{cases} x i x^{-1} &= a\\ x j x^{-1} &= b\\ x k x^{-1} &= c \end{cases}$$ We have \begin{array}{lcl} i^2 = -1 &\implies& a^2 = ( x i x^{-1})^2 = x i^2 x^{-1} = x(-1)x^{-1} = -1\\ ij = k &\implies& ab = (x i x^{-1})(x j ...