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The rift between US President Donald Trump and his former adviser Elon Musk has erupted into the open, with each trading insults after the tech billionaire criticised one of Trump's key domestic policies.

The two billionaires escalated the feud throughout Thursday, lobbing barbs at each other on the social media sites they each own, suggesting a bitter conclusion to their unlikely alliance.

The day began with Trump saying he was "disappointed" with Musk's criticisms of his administration's centrepiece tax and spending bill, musing that it may be the end of their "great relationship".

Musk then accused Trump of "ingratitude", adding: "Without me, Trump would have lost the election".