A square-ordered field, also called a Hilbert field of type (A), is understood to be an ordered field all of whose positive elements are squares. The problem of classifying, up to isomorphism, all
$4$
-dimensional quadratic division algebras over a square-ordered field
$k$
is shown to be equivalent to the problem of finding normal forms for all pairs
$(X, Y)$
of
$3 \times 3$
matrices over
$k, X$
being antisymmetric and
$Y$
being positive definite, under simultaneous conjugation by
${\rm SO}_3(k)$
. A solution is derived for the subproblem of this matrix pair problem defined by requiring
$Y+Y^t$
to be orthogonally diagonalizable. The classifying list is given in terms of a
$9$
-parameter family of configurations in
$k^3$
, formed by a pair of points and an ellipsoid in normal position.
Each
$4$
-dimensional quadratic division algebra
$A$
over a square-ordered field
$k$
is shown to determine, uniquely up to sign, a self-adjoint linear endomorphism
$\alpha$
of its purely imaginary hyperplane. Calling A diagonalizable in case
$\alpha$
is orthogonally diagonalizable, the achieved solution of the matrix pair subproblem yields a full classification of all diagonalizable
$4$
-dimensional quadratic division
$k$
-algebras. This generalizes earlier results of both Hefendehl-Hebeker who classified, over Hilbert fields, those
$4$
-dimensional quadratic division algebras having infinite automorphism group, and Dieterich, who achieved a full classification of all real
$4$
-dimensional quadratic division algebras.
Finally, the paper describes explicitly how Hefendehl-Hebeker's classifying list, given in terms of a
$4$
-parameter family of pairs of definite
$3 \times 3$
matrices over
$k$
, embeds into the classifying list of configurations. The image of this embedding turns out to coincide with the sublist of the list formed by all non-generic configurations.