In AD 4 Augustus was faced with the death of both his heirs, the sons of Julia 1, and the lack of a credible regent. He revised his position.
Julia still had one son left -her youngest, Postumus Agrippa. So Augustus adopted this boy, whom he had previously ignored. The sources suggest there was some deep-seated objection to Postumus, possibly because he had some sort of disability or mental illness. Postumus did not remain an heir for long – he was banished in AD 6.
But Augustus also made a much more interesting move. While focusing his hopes on Gaius and Lucius, Augustus had kept on devising back-up plans involving brutal political marriages and divorces. Let’s go back in time. When Agrippa died in 12 BC, Julia 1 was still young enough to become the mother of even more heirs. Augustus hoped not to need them – he had Gaius and Lucius, her sons. But on the GIRLS ARE MUMS principle Augustus felt Julia should continue to ensure his succession with back-up sons, and picked another suitable husband.
So in 11 BC, Julia married her stepbrother, Livia’s elder son, Tiberius. He was indeed strong enough to enforce the NO MORE WAR principle, but his choice over other candidates was obviously to please Livia. AUGUSTUS LOVES LIVIA. To make this marriage, Tiberius was made to divorce his beloved wife, the mother of his son, and marry Julia against his will. The marriage with Julia was a disaster and produced no children. Julia’s career as mother of heirs was over. By 6 BC, Julia and Tiberius had separated and in 2 BC she was disgraced and sent into exile. Augustus would have to make the best of her children by Agrippa. At the time, with Gaius and Lucius shaping up excellently, this didn’t seem much of a problem.
But in AD 4, the situation was much more urgent. Augustus did something much more radical than just trying out the unsuitable Postumus as an heir – he adopted his stepson, Julia’s ex-husband, Livia’s elder son Tiberius, bringing the Claudius family into the succession. Like Agrippa, Tiberius was no biological relation, but an experienced commander and able to uphold the NO MORE WAR principle. This helps explain how he got chosen to marry Julia. But that doesn’t explain his adoption in his own right, after it was clear that he would not father Julia’s sons – for that we need the fourth principle; AUGUSTUS LOVES LIVIA. Augustus was elevating Livia’s son to a position where he might inherit the rule of Rome – something he had not done for either of Julia’s previous husbands. In fact (spoilers), on Augustus’ death in 14 AD, Tiberius became the Emperor Tiberius.
However, Augustus did not let go of the idea of keeping his own biological family in the line of succession. Initially, he kept his grandson Postumus in reserve; given the age difference, it is probable that he considered placing him next in the succession after Tiberius, so that Tiberius would hold power in his life, but pass it back to a descendant of Augustus at death. Even after he finally gave up on Postumus, Augustus would continue to interfere with the line of succession, diverting it away from Tiberius’ natural heirs towards his own biological family.