find the quantarions that commute with both i and j

Question

anonymous · Answer

i think only real numbers but i would not bet more than $12 on it

anonymous · Answer

$$jk=i\
kj = -i$$ so $k$ does not commute with j

kainui · Answer

I guess any quaternion that works, I'll call it $q$ will have to satisfy this:
$$qi=iq$$$$qj=jq$$

If you take, say the first one, and multiply both sides on the right by j you get this since q commutes with i and j:
$$qij=ijq$$
So since ij=k we now know it commutes with k as well. Interesting.
$$qk=kq$$
Also, it should be mentioned that all numbers commute with real numbers. So q commutes with any real number, it'll be important in a bit.

So now we have 3 commutation relations, and all of them are just equations and we can multiply them by any arbitrary real numbers we want. I'll call the arbitrary real numbers $a$, $b$, $c$, and $d$. 
$$qai=aiq$$$$qbj=bjq$$$$qck=ckq$$$$qd=dq$$
Don't get lost, make sure everything up to this point makes sense cause I'm not explicitly writing down every step. Alright moving on.

We can now add all of these equations together too:
$$qai+qbj+qck+qd=aiq+bjq+ckq+dq$$
factor out q,
$$q(ai+bj+ck+d)=(ai+bj+ck+d)q$$

Yeah, I just showed that q commutes with ALL possible quaternions you could choose. But commuting with all quaternions means  it's a real number. :O

kainui · Answer

This makes me curious cause I never thought about it before. If a quaternion ONLY commutes with i, what possible quaternions could it be?

anonymous · Answer

If a quaternion ONLY commutes with i, I think will be on the form a+bi

kainui · Answer

Yeah I think so too. I was trying to interpret this geometrically, I think if two quaternions commute it means they lie in the same plane since multiplication and rotations within a plane is all commutative.

anonymous · Answer

thanks for your help!