Groote Eylandt is the largest island in the Gulf of Carpentaria and the fourth largest island in Australia.It is the homeland of, and is owned by, the Warnindhilyagwa who speak the Anindilyakwa language.. Groote Eylandt lies about 50 km (31 mi) from the Northern Territory mainland and eastern coast of Arnhem Land, about 630 kilometres (390 mi) from Darwin, opposite Blue Mud Bay.