Difference between properties of 3D unit vectors and 4D unit vectors(in the context of quaternions) In the book I'm reading about quaternions, the properties of operations with imaginary values $i,j,k$ are compared to properties of the cross product of Cartesian vectors $\mathbf{i}, \mathbf{j}, \mathbf{k}$. E.g. $$ij=k\\\mathbf{i} \times \mathbf{j}=\mathbf{k}.$$ So then we have $$\mathbf{i}\times\mathbf{j}=\mathbf{k}\\ \mathbf{i}\times\mathbf{k}=\mathbf{-j}.$$ And then the author assumes that these vectors obey distributive and assocaitive axioms to prove their imaginary properties: $$\mathbf{iij}=\mathbf{ik}=\mathbf{-j} \\ \mathbf{ii}=\mathbf{i}^2=-1$$ But what I don't understand is why these vectors would obey associative axiom, since $\mathbf{i}\mathbf{i}\mathbf{j}$ are still cross products, or at least I assume so. And what about $\mathbf{i}\times\mathbf{i}=0$? Does this mean that these are not actually Cartesian vectors? I suppose, that I am missing some key points in this explanation...
 A: Quaternions have only 4 letters to write (1,i,j,k), so you can check by hand that multiplication is assiociative.
However, multiplication of quaternions is not commutative.
Problem with 3 dimensions, 5 dimensions, 7 dimensions etc is that there is no way to define a field structure on 3/5/7/9...etc dimensional real space, even if we don't  require commutavitity.
On 1 dimension we have natural field structure, on 2 dimensions the only possible field structure is complex numbers.
On 4/6/8/10...etc dimensions, we can define structure of field but only noncommutative. When dim=4, these are quaternions.
It is not true that i*i=0, I don't understand what do you mean.
A: You are confusing cross product with quaternion product. The notation $i \times j = k$ is referring to cross product, while the notation $i j = k$ is referring to (pure) quaternion product. Let me explain the later:
Multiplication of two pure quaternions $A$ and $B$ can be written in vector form as:
$$A B = - A \cdot B + A \times B$$
Where a dot product is involved as well as cross prpduct. You can check that by calculanting $A B$ using the quaternion multiplication rules and grouping terms.
Using that product you can check that $i i = -1$ and that $i k = -j$ and also that it is associative is associative like $i i j = -j$
