= Two-generator HNN embedding
{title2=$G\hookrightarrow E=\langle a,t\rangle$}
For $G=\langle g_i\mid R\rangle$, put $B=G*F(a,b)$, $u_0=a$, $v_0=b$, $u_i=b^{-i}ab^i$ and $v_i=g_i a^{-i}ba^i$ for $i\ge1$. 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 $t^{-1}u_it=v_i$ embeds $G$ by <Britton's lemma>. Since $b=t^{-1}at$ and $g_i=t^{-1}u_it(a^{-i}ba^i)^{-1}$, 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>.
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