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Cofibrant CDGAs are retracts of standard ones. A standard cofibrant CDGA is a free commutative graded algebra on a (possibly transfinite) sequence of generators $x_1,x_2,\dots$ such that $d(x_i)$ only depends on previous generators. A linear basis is given by monomials $x_{i_1}^{n_{i_1}}\cdots x_{i_r}^{n_{i_r}}$ such that $i_j<i_{j+1}$ and $n_{i_j}=1$ if $|x_{i_j}|$ is odd. You can put a kind of lexicographic order in these monoimials in such a way that the differential of each one only depends on previous monomials. This is cofibrant as a complex in the projective model structure.