Why is this algebraic manipulation of quaternions incorrect? I know that for quaternions,
$$i^2=j^2=k^2=ijk=-1$$
I've tried to understand this intuitively as having $i$, $j$ and $k$ represent a rotation about each of three axes. But when I do a bit of manipulation with the quaternions, I ran into a problem. If
$$i^2=ijk$$
shouldn't
$$i=jk \text{ ?}$$
But because
$$i^2=-1$$
$$(jk)^2=-1$$
and
$$j^2k^2=-1$$
which is obviously not true: $j^2k^2$ should equal $1$.
Where is my error?
By the way, Wikipedia notes that quaternionic multiplication is non-commutative. Does that have something to do with it?
 A: I view quaternions as reading. The order is $\{1, i, j, k\}$ so we read left to right.  That is, $ij = k$ since that is left to right whereas $ji = -k$ since this is right to left so it is reversed, $-1$.
$$
(jk)^2 = jkjk = ii = -1
$$
Also note that 
$$
j^2k^2 = jjkk = jik = -kk = 1\neq jkjk = ii = -1
$$
Does this help?
A: There is another way of looking at quaternions - they are identical to matrices of a certain kind. Quaternions of the form $bi + cj + dk + a$ can be represented by matrices of the form
$$
\begin{bmatrix}
a+bi &  c+di \\
-c+di & a-bi
\end{bmatrix}
$$
where the $i$ inside the matrix is $\sqrt{-1}$.
The matrix representing the quaternion $i$ is 
$$
\begin{bmatrix}
i &  0 \\
0 & -i
\end{bmatrix}
$$
Write down $jk$ in this form
$$
jk=
\begin{bmatrix}
0 &  1 \\
-1 & 0
\end{bmatrix}
\begin{bmatrix}
0 &  i \\
i & 0
\end{bmatrix}
=
\begin{bmatrix}
i &  0 \\
0 & -i
\end{bmatrix}
$$
which is clearly $i$, not $-i$.
The rule is do nothing with quaternions that you cannot do with matrices of the form given above and you'll be ok.
Edited to add: If you think of $(jk)^2$ in terms of matrix multiplication, 
$$(jk)^2=jkjk=ii=
\begin{bmatrix}
i &  0 \\
0 & -i
\end{bmatrix}
\begin{bmatrix}
i &  0 \\
0 & -i
\end{bmatrix}
=
\begin{bmatrix}
-1 &  0 \\
0 & -1
\end{bmatrix}
=-1
$$
But $j^2k^2$ = $jjkk = jik = $
$$
\begin{bmatrix}
0 &  1 \\
-1 & 0
\end{bmatrix}
\begin{bmatrix}
i &  0 \\
0 & -i
\end{bmatrix}
\begin{bmatrix}
0 &  i \\
i & 0
\end{bmatrix}
=
\begin{bmatrix}
1 &  0 \\
0 & 1
\end{bmatrix}
=1
$$
