So the following relations are suppose to follow from :



But, for instance, multiplying through by in the relation yields:



This is my problem, how can . I suspect that I've missed some vital detail, perhaps some subtly to do with the commutation law.

Multiplication in quatenionic algebra should be considered an "operation" for conceptual clarity.(e.g multiplication by iota in the complex plane signifies an anticlockwise rotation by a right angle)
Suppose we have

When the operator i acts on ij the result will be the same as i acting on -ji.That is,
(this is called pre multiplying by i)
Now we know,

Pre multiplying b/s by ij we have
or
(because ij and -ji are the same operators)
This is called post-multiplication by i.
You have pre-multiplied by i on the LHS but post-multiplied on the RHS which has led to this error.

Thank you Sameed. The ordering of symbols is essential. Viewing these entities as "operators" makes this clearer. I have many, many more questions; too many to ask here though - I'll have to save up to buy a book on quaternions (and octonions).