Let $F_2$ denote the free group of rank 2. Does anybody have a fast proof that the subgroup membership problem is undecidable for $F_2 \times F_2$? I saw a really fast proof last semester that started with a group with undecidable word problem and used that group to construct subgroups of $F_2 \times F_2$, but I've lost my notes from the talk. I tried looking at the original proof from the 1960's but I couldn't get an English translation. Thanks!