Connection between quaternions and electromagnetism Consider the multiplication of two quaternions, which can be written as a matrix equation:
$$
xy = \begin{pmatrix}
  x_0 & -x_1 & -x_2 & -x_3 \\
  x_1 & x_0 & -x_3 & x_2 \\
  x_2 & x_3 & x_0 & -x_1 \\
  x_3 & -x_2 & x_1 & x_0
 \end{pmatrix}\begin{pmatrix}
  y_0\\y_1\\y_2\\y_3
 \end{pmatrix} = x_0y + \begin{pmatrix}
  0 & -x_1 & -x_2 & -x_3 \\
  x_1 & 0 & -x_3 & x_2 \\
  x_2 & x_3 & 0 & -x_1 \\
  x_3 & -x_2 & x_1 & 0
 \end{pmatrix}\begin{pmatrix}
  y_0\\y_1\\y_2\\y_3
 \end{pmatrix}
$$
if we put the four components of a quaternion inside a four-component vector. Let's compare this to the electromagnetic field strength tensor, which is given in terms of the electric and magnetic fields as:
$$
F^{\mu\nu} = \begin{pmatrix}
     0     & -E_x & -E_y & -E_z \\
     E_x &  0     & -B_z   &  B_y    \\
     E_y &  B_z   &  0     & -B_x   \\
     E_z & -B_y   &  B_x   &  0
  \end{pmatrix}
$$
The structure of these matrices seems oddly similar. Why could this be?
I assume this has to do with $F^{\mu\nu}$ being a differential 2-form, at least that's where I think the antisymmetry comes from.
 A: The field strength tensor can be written as
$$\tag{1}
F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu=2\,\partial_{[\mu}A_{\nu]}
$$
where $A_\mu$ is the vector potential. This is antisymmetric in all four indices $\mu=0,1,2,3$ (and $\nu=0,1,2,3$ of course).
The multiplication of two quaternions
\begin{align}
p&=x_0+\boldsymbol{i}x_1+\boldsymbol{j}x_2+\boldsymbol{k}x_3\,,\\
q&=y_0+\boldsymbol{i}y_1+\boldsymbol{j}y_2+\boldsymbol{k}y_3\,
\end{align}
can be written as
\begin{align}
pq&=\underbrace{x_0y_0-x_1y_1-x_2y_2-x_3y_3}_{\text{scalar}}\\
&+\underbrace{\boldsymbol{i}(x_0y_1+x_1y_0)+\boldsymbol{j}(x_0y_2+x_2y_0)+\boldsymbol{k}(x_0y_3+x_3y_0)}_{\text{symmetric}}\\
&+\underbrace{\boldsymbol{i}(x_2y_3-x_3y_2)+\boldsymbol{j}(x_3y_1-x_1y_3)+\boldsymbol{k}(x_1y_2-x_2y_1)}_{\text{anti symmetric}}\,.
\end{align}
Even if we take from this only the antisymmetric part it is only in three indices $1,2,3$ which makes it pretty hopless to formulate (1) only with quaternions.
A viable approach though is Space time algebra that uses the Clifford algebra $\mathcal{Cl}_{1,3}(\mathbb R)$ which is built on Dirac matrices. In a sense this is a space of higher "quaternions" that has the right signature for special relativity.
