In group theory and especially in the group $GL(n,F)$ we define for $i\neq j $ $X_{ij}(a) $to be the $n\times n$ matrix whose $(k,l)$ entry is equal to a if $(k,l)=(i,j)$ and equal to $\delta_{kl}$ otherwise . $X_{ij}$ is a subgroup of $GL(n,F)$ called the root subgroup. The matrices $X_{ij}(a)$ and their conjugates by elements of G are called transvection.
My question is: Is the product of transvections still a trasvection? Or simpler still, is $X_{ij}(a)X_{i'j'}(b)$ or generally $\prod_{i,j}X_{i,j}$ a transvection? If that doesn't hold, is there a weaker form of it that s true, for example if i=i'? I know for a fact that the root subgroups are conjugate to each other in $SL(n,F)$ but that doesn't seem to help much.