  Avramov, Conca, and Iyengar ask whether $\beta_i^S(R) \leq \binom{g}{i}$ for
all $i$ when $R=S/I$ is a Koszul algebra minimally defined by $g$ quadrics. In
recent work, we give an affirmative answer to this question when $g \leq 4$ by
completely classifying the possible Betti tables of Koszul algebras defined by
height-two ideals of four quadrics. Continuing this work, the current paper
proves a structure theorem for Koszul algebras defined by four quadrics. We
show that all these Koszul algebras are LG-quadratic, proving that an example
of Conca of a Koszul algebra that is not LG-quadratic is minimal in terms of
number of defining equations. We then characterize precisely when these rings
are absolutely Koszul, and establish the equivalence of the absolutely Koszul
and Backelin--Roos properties up to field extensions for such rings (in
characteristic zero). The combination of the above paper with the current one
provides a fairly complete picture of all Koszul algebras defined by $g \leq 4$
quadrics.
