"I asked that Antoninus should also adopt Lucius's son, who will in this way become your brother"

This refers to Lucius Verus, the son of Hadrian's adopted son, Lucius Aelius. Although Marcus Aurelius seems not to have been close to Hadrian (he pointedly does not include him among those listed, at the beginning of his Meditations, as having influenced the development of his own character),  he did, on his accession, insist on the full implementation of the terms of Hadrian's will, refusing to accept power unless Lucius Verus could rule alongside him, which he did until his death in 169 AD. It was the first, but not the last time that the Empire was ruled by two emperors. The two men held equal status in law, although Marcus Aurelius was always seen as the senior partner, and alone held the religious office of Pontifex Maximus. Lucius Verus put down a rebellion in Syria in 162-3 AD, and was awarded a Triumph on his return. He fell ill in 168 AD, possibly from smallpox, or possibly from the "Antonine Plague," an epidemic which ravaged Rome several times in the 2nd Century AD. He was deified following his death.