Let $k$ be a field of characteristic zero, and let $A_1=A_1(k)$ be the first Weyl algebra.

It is well known (first proved by Dixmier, if I am not wrong) that the group of automorphisms of $A_1$, denote it by $H$, is the free amalgamated product of two of its subgroups (the affine and the triangular automorphisms), see for example, [Theorem 2][1].

Denote the group of automorphisms and anti-automorphisms of $A_1$ by $G$. 
It is easy to see that $H$ is a normal subgroup of $G$ (of index $2$).

**My question:** Is it true that $G$ is the free amalgamated product of two of its subgroups, namely, the affine and the triangular automorphisms and anti-automorphisms? 

It seems to me that my question should have a positive answer,
with a proof for $G$ similar to a proof for $H$ (am I missing something?).

Thank you very much for your help.

**Edit:** I do not know if [Karrass and Solitar's paper][2] may help answer my question, since it deals with subgroups of a free amalgamated group,
while in my case a free amalgamated group appears as an index $2$ (normal) subgroup of another group (can one prove that a group is free amalgamated by knowing that it has an index $2$ subgroup which is free amalgamated?).

  [1]: http://archive.numdam.org/article/BSMF_1984__112__359_0.pdf
  [2]: http://www.ams.org/journals/tran/1970-150-01/S0002-9947-1970-0260879-9/S0002-9947-1970-0260879-9.pdf