Is the quaternion division well defined?

Question

Given that in quaternions generally $pq^{-1}\ne q^{-1}p$, how can quaternions form a division algebra?

For instance, $( i+2 j)\cdot\frac{1}{5 j+1}=\frac{1}{26} (i+2 j-5 k+10)$ but $\frac{1}{5 j+1}\cdot( i+2 j)=\frac{1}{26} (i+2 j+5 k+10)$, which makes $\frac{ i+2 j}{5 j+1}$ undefined and ambiguous, yes?

The Wikipedia article in quaternion says "Because it is possible to divide quaternions, they form a division algebra." But the example above seemingly shows that it is imnpossible to unambiguously define division of quaternions. Are there mistakes in the example?

UPDATE

I found this page where they define division of quaternions the following way:

$\frac{a_0+a_1i+a_2j+a_3k}{b_0+b_1i+b_2j+b_3k}=\frac{a_0b_0+a_1b_1+a_2b_2+a_3b3}{b_0^2+b_1^2+b_2^2+b_3^2}+i\frac{a_1b_0-a_0b_1-a_3b_2+a_2b3}{b_0^2+b_1^2+b_2^2+b_3^2}+j\frac{a_2b_0+a_3b_1-a_0b_2-a_1b3}{b_0^2+b_1^2+b_2^2+b_3^2}+k\frac{a_3b_0-a_2b_1+a_1b_2-a_0b3}{b_0^2+b_1^2+b_2^2+b_3^2}$

So, my question is, how can they do it if quaternion division is ambiguous as the example above shows?

Answer 1

This may be mostly about terminology.

In quaternions, division is well -defined in the sense that for any $p$ and nonzero $q$, the expressions $p q^{-1}$ and $q^{−1}p$ exist.

On the other hand, since in general $p q^{-1}\neq q^{−1}p$, the expression $p/q$ (and the wordy expression "$p$ divided by $q$") are ambiguous and thus probably best avoided. You can of course pick some convention, but that will most likely lead to misunderstandings.

Note that the situation is the same with (invertible) matrices or operators.