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How do you prove that these relations are correct $(ij = k, jk = i, \ldots)$?

I tried to prove some of them, and I could, but for example:

ik = -j
-j = -1 * j = (ijk) * j  = i*(j^2)*k = -1 (ik) = -ik
so -j = -ik... 

which is wrong, ** in which step(s) am I making the mistake?** (I am sure the error is commutativity)

Also it can proven like this (or so I have found):

ik = -j
ik = (i)^-1 * k^(-1) = (ki)^-1 = (j^-1) = -j
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Your step going from $(ijk)\cdot j = i\cdot(j^2)\cdot k$ is wrong. You appear to have used the commutativity of multiplication to exchange the order of $k$ and $j$. But quaternion multiplication is not commutative.

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You have $a^2 = -1$ for $a \in \{i,j,k\}$ and $ijk = -1$.

To show $ik = -j$, start with $ijk = -1$ and multiply both sides on the left by $i$ and on the right by $k$, then using the identities above you end up with $(-1)j(-1) = j = i(-1)k = -ik$.

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