Looking at ijk = -1, i'm trying to fully understand how to find individual values.

i appears easy: i(ijk) = -1(i) -> i²jk = -i -> -jk = -i -> jk = i

Trying to find j is problematic: j(ijk) = -1(j) -> ij²k = -j -> -ik = -j -> ik = j

That's wrong, it should be ki = j and ik = -j

Can someone point out what i'm doing wrong here?

It seems that the ij²k = -j -> -ik = -j step is wrong, but i don't know how to handle i-k = -j

Am i just free to arbitrarily re-arrange it to -ki = -j, that seems wrong somehow....

34 minutes ago, mathematical said:

j(ijk) = -1(j) -> ij²k = -j

There's your error.  You can't move from j(ijk) to ij²k.  j(ijk) = jijk = (ji)jk and ij²k = ijjk = (ij)jk, which are not equivalent for quaternions.  ij = k and ji = -k, so j(ijk) = -ij²k.

That makes sense, so:

ijk = -1 -> j(ijk) = -j -> jijk = -j -> (ji)jk = -j -> jk = -j²i -> k = -ji -> ji = -k

What i'm not 100% confident in is how i go from j(ijk) to -ij²k, this is what i think:

ijk = -1 -> j(ijk)= -j -> j = -j(ijk) -> j = -ij²k

Is that right or am i missing a step? When i move (ijk) over, the negative term applies to everything, not just j, right?

I'm not sure what axioms you are assuming in your computations. In other words, I don't know which known identities you are using to try to deduce other things.

To me the basic identities are:

i² = j² = k² = -1

ij = -ji = k

jk = -kj = i

ki = -ik = j

You can easily compute anything with those. You seem to start with ijk=-1, which is true, but I had to do (ij)k = k² = -1 in my head to verify it.

To go from j(ijk) to -ij²k, I simply swap the order of the first "ji" to "ij", which makes the sign flip: j(ijk) = (ji)jk = -(ij)jk = -ij²k

Alternatively, just compute both sides and verify they are -j.

I'm trying to understand the steps needed to prove that ij = -ji = k, etc....

Like i get that i can flip ji to ij and make the sign flip, but i don't get why (well, i kind of do now)

Trying to really zero in on the "how does this work" bits

Edit, this post was dumb, removing it and going to sleep.

1 hour ago, mathematical said:

I'm trying to understand the steps needed to prove that ij = -ji = k, etc....

I've always thought of quaternions with all those identities to begin with, but there's probably a shorter list of properties from which all of these can be deduced. I imagine that's what you are trying to do. That's why I was asking what axioms you are starting from.

So far i can prove that \( i = jk \) and that \(k = ij \), i can't prove that \( j = ki \). Also, can't sleep thinking about this. Here is what i have:

• \( ijk = -1 \rightarrow i(ijk) = -i \rightarrow i^{2}jk = -i \rightarrow -jk = -i \rightarrow \mathbf{jk = i} \)
• j = ki?
• \( ijk = -1 \rightarrow k(ijk) = -k \rightarrow (kij)k = -k \rightarrow  k = -k(kij) \rightarrow k = -k^{2}ij \rightarrow \mathbf{k = ij} \)

I think i got it....

\( j(ijk) = -j \rightarrow ji(jk) = -j \rightarrow ji = -j(jk) \rightarrow ji = -j^{2}k \rightarrow ji^{2} = -j^{2}ki \rightarrow -j = -j^{2}ki \rightarrow -j = -ki \rightarrow \mathbf{j = ki} \)

Does that look about right? I think going from ji = -j²k to ji² = -j²ki should be good, this is the step i was missing. But after looking at this for so many hours, i'm not 100% confident...

You still haven't given me the list of axioms you are starting with, and without that I can't even start to help you.