  Let $\kk$ be a field, $R$ a standard graded quadratic $\kk$-algebra with
$\dim_{\kk}R_2\le 3$, and let $\ov\kk$ denote an algebraic closure of $\kk$. We
construct a graded surjective Golod homomorphism $\varphi \colon P\to
R\otimes_{\kk}\ov{\kk}$ such that $P$ is a complete intersection of codimension
at most $3$. Furthermore, we show that $R$ is absolutely Koszul (that is, every
finitely generated $R$-module has finite linearity defect) if and only if $R$
is Koszul if and only if $R$ is not a trivial fiber extension of a standard
graded $\kk$-algebra with Hilbert series $(1+2t-2t^3)(1-t)^{-1}$. In
particular, we recover earlier results on the Koszul property of Backelin,
Conca and D'Al\`i.
