Everywhere that I've looked, it seems to be assumed that $i^{2} = j^{2} = k^{2} = - 1$, along with the other rules of quaternion multiplication. However - for my homework - I'm being asked to show these rules are valid. Can someone point me in the right direction for going about doing this?

I've been given are the values of $I, J, K$ on the standard basis of $R^{4}$ (i.e. $I(e_1)=e_2)$. Using these rules I see how to show that it's a group and how to complete the rest of the question - I just don't know how to prove that $i^{2}$, etc. are calculated.


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    $\begingroup$ What definition of the quaternions are you working with? Ordinarily I would take $i^2=j^2=k^2=ijk=-1$ as the definition and prove that the resulting vector space has division ring structure. $\endgroup$ – user7530 Jan 21 '13 at 1:00
  • $\begingroup$ What is the exact definition of quaternions that you are given? $\endgroup$ – Shaun Ault Jan 21 '13 at 1:00
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    $\begingroup$ Can you use your rules to show that $I(I(e_i)) = -e_i$ for every $i$? $\endgroup$ – user7530 Jan 21 '13 at 1:24
  • $\begingroup$ That's a fantastic idea! It looks like it'll help with all of the multiplication. Thanks for your help! $\endgroup$ – Luke8ball Jan 21 '13 at 1:32

Hint: The method suggested by @user7530 in the comments is a good one. Here's another way to go. With this definition the quaternions are matrices. Find the matrix representations of $i$, $j$, and $k$ and simply multiply the matrices to find the required result, keeping in mind that the matrix representation of $1$ is the unit matrix.


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