Magnetism is relativity in disguise
A line of static charge produces only an electric field. Run alongside it and two things happen at once: the charges now constitute a current, giving a magnetic field; and length contraction alters the charge density, changing the electric field too. \mathbf{E} and \mathbf{B} are not independent absolutes — they are components of one object that reshuffle when you change frames. Relativity is not bolted onto electromagnetism; Maxwell's equations were relativistic all along, with c built in from the start.
Four-current and four-potential
Package charge density and current into the four-current J^\mu=(c\rho,\ \mathbf{J}). Then charge conservation — the continuity equation — becomes one clean statement: the four-divergence \partial_\mu J^\mu=0, manifestly identical in every inertial frame.
The four-current and charge conservation as a Lorentz-invariant four-divergence.
Likewise the scalar and vector potentials join into the four-potential A^\mu=(\phi/c,\ \mathbf{A}), and the Lorenz gauge condition becomes the equally clean \partial_\mu A^\mu=0. In that gauge the sourced wave equation of guide 2 is one covariant line: \Box A^\mu=\mu_0 J^\mu.
The four-potential, the Lorenz condition, and the wave equation of electromagnetism in a single covariant line (\Box=\partial_\mu\partial^\mu).
The field tensor
The fields themselves live in the antisymmetric electromagnetic field tensor F^{\mu\nu}=\partial^\mu A^\nu-\partial^\nu A^\mu. An antisymmetric 4\times4 matrix has exactly six independent entries — precisely the three components of \mathbf{E} and the three of \mathbf{B}, now bound into one geometric object.
The field tensor built from the four-potential; the time-space entries hold \mathbf{E}/c, the space-space block holds \mathbf{B}.
There is also a dual field tensor *F^{\mu\nu}=\tfrac12\varepsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}, built with the four-index Levi-Civita symbol. It is what you get by swapping \mathbf{E}/c\leftrightarrow\mathbf{B} (with a sign) — the mathematical shadow of the near-symmetry between electricity and magnetism.
Maxwell's equations in two lines
Now the payoff. All four of Maxwell's equations collapse into two tensor equations. The inhomogeneous pair (Gauss's law and the Ampère-Maxwell law, displacement current included) sits in \partial_\mu F^{\mu\nu}=\mu_0 J^\nu; the homogeneous pair (no magnetic monopoles and Faraday's law) is the cyclic identity below.
Maxwell's equations, complete, in covariant form; the second (equivalently \partial_\mu {*F}^{\mu\nu}=0) holds automatically because F comes from a potential.
How E and B mix under a boost
Because \mathbf{E} and \mathbf{B} are one tensor, a Lorentz boost rotates them into each other. Boosting with velocity \mathbf{v} (and Lorentz factor \gamma), the field components parallel to the boost are unchanged, while the perpendicular ones mix electric and magnetic.
Transformation of the fields under a boost: parallel components fixed, perpendicular ones mixed.
Two combinations never change — the Lorentz invariants \mathbf{E}\cdot\mathbf{B} and E^2-c^2B^2. So if \mathbf{E}\perp\mathbf{B} in one frame (as in any radiation wave) it is so in all frames; and a field that is purely electric somewhere can never be transformed to purely magnetic unless the sign of E^2-c^2B^2 permits it. The invariants are the fixed bones beneath the frame-dependent flesh.
The two Lorentz invariants of the electromagnetic field.