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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 $\kk$. Weconstruct a graded surjective Golod homomorphism $\varphi \colon P\toR\otimes_{\kk}\ov{\kk}$ such that $P$ is a complete intersection of codimensionat most $3$. Furthermore, we show that $R$ is absolutely Koszul (that is, everyfinitely 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 standardgraded $\kk$-algebra with Hilbert series $(1+2t-2t^3)(1-t)^{-1}$. Inparticular, we recover earlier results on the Koszul property of Backelin,Conca and D'Al\`i.

The absolutely Koszul and Backelin-Roos properties for spaces of quadrics of small codimension