How can I define $i^2=j^2=k^2=-1$ in Mathematica?

Question

First, I want to define the identities as

$i^2=j^2=k^2=-1$,

$ij=k=-ji$,

$jk=i=-kj$,

$ki=j=-ik$.

And then I want to use these identities in my sequence

$Q_n = F_n + iF_{n+1} + j F_{n+2} + k F_{n+3}$,

where $F_n$ is the Fibonacci sequence and $n \geq 0$.

For example I want to calculate and simplify all results for $Q_{11} Q_{9} - Q_{10}^2 = ?$.

With my best.

Answer 1

You can use the Quaternions package. Load:

<<Quaternions`

Define your sequence:

Q[n_] := Fibonacci[n] + I Fibonacci[n+1] + J Fibonacci[n+2] + K Fibonacci[n+3]

Then your example:

Q[11]**Q[9] - Q[10]**Q[10]

2 + 2 J + 5 K