The exponential function for the imaginary part of the quaternion is defined like this: $\exp(q) = \sin(|q|)+\cos(|q|)\times q|q|$
When I set $q=q_1+q_2$, then $\exp(q)=\exp(q_1+q_2)=\exp(q_1)\exp(q_2)$. Is this already a mistake here?
When I then multiply it out with some quaternions $q_1=i$ and $q_2=j$, then $\exp(i+j)=\sin(\sqrt2)+\cos(\sqrt2)(i+j)\sqrt2$, but\begin{align*}\exp(i+j)&=\exp(i)\exp(j)\\&=(\sin(1)+\cos(1)i)(\sin(1)+\cos(1)j)\\&=\sin^2(1)+\sin(1)\cos(1)(i+j)-\cos^2(1)k,\end{align*}which aren't the same.
Why are these results different? Where is the mistake and why does it occur?
This question also applies to geometric algebra, but since this seems to be less understood, I ask about quaternions, which have the same problems for the exponential function.
A basic problem is that it is not true, in general, that $\exp(q_1+q_2)=\exp(q_1)\exp(q_2)$. This holds, however, if $q_1$ and $q_2$ commute. But, since $i.j=-j.i$, you have no reason to believe that $\exp(i+j)=\exp(i).\exp(j)$.
By the way, the usual definition of the exponentional function within the quaternions (and elsewhere) is$$\exp(q)=\sum_{n=0}^\infty\frac{q^n}{n!}.$$