Given $a_1,b_1,c_1$ in $F_2[t]$ with $\gcd(a_1,b_1,c_1)=1$ it is known that there exists an element $g$ of $SL(3, F_2[t])$ (by explicit construction) such that $g$ has first line $$ [a_1^2, b_1,c_1], $$ in particular this works when $a_1^2$ is in the ideal $(b_1,c_1)$ generated by $a_1$ and $b_1$ in $F_2[t].$
Question: It is possible to extend in the above manner the following line $L$ to an element of $SL(3, F_2[t])$
$$ L =[1+a,b,c] $$ where $a,b,c \in F_2[t]$ with $\gcd(1+a,b,c)=1$ and with $$ a^2 \in (b,c). $$
The case when $\gcd(b,c)=1$ is trivial since $1+a = M^2+ t N^2$ and $tN^2 \in (b,c)$ in these case.
The question (for a general characteristic $2$ ring R) appears in page $14$ of
while the explicit construction described above appears in page $12.$