The figure below outlines a derivation which
uses frame-invariance
of the proper-force F_{o}
(i.e. the proper-acceleration^{[1]}
times
the rest mass) to derive Biot-Savart (the equation for determining
the magnetic field caused by current in a wire element) from the vantage
point of forces F experienced in a single frame. When these forces
are perpendicular to the line of a test particle's velocity
v_{⊥}, proper-force is simply related to the frame-variant
force F (defined as usual in terms of momentum and energy transfer) by

F_{o} = γ_{⊥}F,
where γ_{⊥} ≡ 1/Sqrt[1-(v_{⊥}/c)^{2}].

Illustrated below are the simplest cases
of electric and magnetic forces from a current element, specifically
(i) a test charge stationary with respect to
a neutral wire carrying an electron current, and (ii) a test charge moving
along with the electron current in that wire. For the first case (top
half of figure), the forces act on a stationary
particle so that v_{⊥} is zero and γ_{⊥} is
1. Thus the forces from positive and negative charges in the neutral
wire cancel out. Magnetic effects are not seen.

For the case at the bottom on the figure, we again only have to
worry about the force perpendicular to the line of particle motion.
In this case, the proper-force exerted on our test charge by *the stationary
positive charges*
remains in the direction of their separation, but
increases in magnitude by that factor of gamma, i.e.
F_{o+}'=γF_{+}.

What *the moving electrons* do to the test particle as it
matches speed with them
would be less clear, except that by symmetry the proper-force in
that case (F_{o-}') must decrease by the same
factor of gamma, since the test particle now moves with
them rather than the protons. In other words,
F_{o-}'=-F_{+}/γ. Thus the original
proper-force from the protons is multiplied by (γ - 1/γ) to
yield the total , justifying *from our perspective*
the moving test charge's sense that the wire has become
positive via length contraction i.e. protons
closer together and electrons further apart. Hence the
net force F' from the lab point of view is the
original frame-variant force from only
the positive charges F_{+}, multiplied
by (γ-1/γ)/γ or simply by
(v/c)^{2}.

One of those v's (hidden in current I) is used to generate the intermediate
quantity B (magnetic field) calculated by Biot-Savart, while the other takes
up residence in
the Lorentz expression for predicting the effect of such magnetic
fields on a moving particle. The magnetic field thus allows one to
conceptualize a field-structure around the wire, whose forces *on any moving
charge* may be predicted by the Lorentz Law.
Biot-Savart and the Lorentz Law, of course,
in vector form also work for arbitrary configurations of moving
charge not covered by this simple derivation. A slight
change
in notation
exploits this relativistic symmetry even further.
The use of proper
force here also makes it clear that similar relativistic or
"magnetic-force-like" corrections will apply to any "action at a distance"
force in flat (3+1)D spacetime.

The derivation above is simpler than the multi-frame derivation below, which however develops the moving charge's perspective more fully. This multiframe derivation is discussed in some detail by Dan Schroeder in his notes here on "Purcell Simplified: Magnetism, Radiation and Relativity".

^{[1]} Proper-acceleration
and proper-force are nothing more than
the 3-vector acceleration and net force experienced in the
frame of an accelerated object. Like proper time and
proper length, they describe quantities
from the vantage point of a special frame, namely
that of the object to which the acceleration and forces
apply. Their magnitude also happens to be the Lorentz-invariant magnitude
of the net force and acceleration four-vectors as seen from
any frame, since in the rest frame of the accelerated object the
time-like component of these four-vectors is zero. Because
forces might be caused by objects moving with respect to one
another, proper forces don't obey an action-reaction principle.
The equivalence principle makes these quantities useful even
in non-inertial settings (like gravitationally-curved spacetime).
The utility of Newton's Laws for describing gravity on
earth, along with other inter-object forces, is living proof
of that.