Two-generator HNN embedding ($G\hookrightarrow E=⟨a,t⟩$)

For $G=⟨g_i\mid R⟩$, put $B=G\ast F(a,b)$, $u_0=a$, $v_0=b$, $u_i=b^{-i}a{b}^i$ and $v_i=g_i{a}^{-i}b{a}^i$ for $i\ge 1$. The $u_i$ freely generate by free conjugate bases in a rank-two free group. The $v_i$ do too: projection killing $G$ maps them to a free conjugate basis, so no reduced word in them is trivial. The HNN extension adjoining $t^{-1}{u}_{i}t=v_i$ embeds $G$ by Britton's lemma. Since $b=t^{-1}at$ and $g_i=t^{-1}{u}_{i}t(a^{-i}b{a}^i{)}^{-1}$, its generators reduce to $a,t$. For finitely many $g_i$ and $R$ this is a finite group presentation; for effective countably many generators and enumerable $R$ it is a two-generator recursive presentation of a group.