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The government has appointed Adam Tolley KC to investigate two complaints made about the conduct of Deputy Prime Minister Dominic Raab.

Mr Raab asked Prime Minister Rishi Sunak to launch an investigation into his own conduct after allegations about his behaviour towards staff.

Downing Street said Mr Tolley's report will be published "in a timely way".

Any final judgement on whether Mr Raab was in breach of the ministerial code will remain with Mr Sunak.

"The prime minister will remain the arbiter of the code," the PM's official spokesman told reporters.

The Liberal Democrats said Mr Sunak "must commit now to sacking Dominic Raab if complaints about his bullying are upheld".

The party's Chief Whip Wendy Chamberlain added: "Anything less would make a mockery of his promise to bring back integrity."

Labour deputy leader Angela Rayner warned the investigation must not be a "whitewash" and called on Mr Sunak to expand its scope to cover other allegations about Mr Raab's behaviour.

"This Conservative government has a troubling track record of brushing serious misconduct under the carpet. Their refusal to act on findings against [former home secretary] Priti Patel previously led their former ethics chief to quit in disgust," said Ms Rayner.

"A temporary stop-gap investigator, appointed in a panic, with an absurdly narrow remit is not a solution to dealing with the flood of allegations of ministerial misconduct now requiring investigation."

Mr Tolley specialises in commercial and employment law. He represented HM Revenue and Customs in a High Court case against three ´óÏó´«Ã½ News presenters, which resulted in them having to pay back tens of thousands of pounds in tax.

As an independent investigator, he has been asked by the government to "establish the specific facts" about two formal complaints against Mr Raab.

The complaints relate to Mr Raab's conduct when he was foreign secretary and later justice secretary under Boris Johnson.

Mr Raab was sacked as justice secretary and deputy prime minister by former PM Liz Truss when she took power in September.

But the Esher and Walton MP was reappointed to both roles by Mr Sunak following his election as Conservative leader by the party's MPs.

The complaints about his past behaviour , which said it had spoken to multiple sources who claimed Mr Raab had created a "culture of fear" at the Ministry of Justice.

Concerns have also been expressed to ´óÏó´«Ã½ News about Mr Raab's behaviour in meetings.

Lord McDonald, the former top civil servant at the Foreign Office, then went public with claims that many colleagues were "scared" to go into Mr Raab's office when he was foreign secretary.

"It was language, it was tone, he was very curt with people," Lord McDonald told Times Radio, "and he did this in front of other people, people felt demeaned."

The government promised Mr Tolley "will have access to all the information he wishes to see" and that his report on Mr Raab's conduct will be made public.

Such a probe would normally fall within the remit of the PM's adviser on ministerial conduct.

But that position has been vacant since the previous occupant Lord Geidt quit in June after conceding Mr Johnson may have broken ministerial rules over Partygate.

Mr Sunak has said he is in the process of recruiting a new ethics adviser.

The two formal complaints were lodged against Mr Raab on 15 November.

The following day, Mr Raab wrote to Rishi Sunak to ask for an inquiry, hours before he has due to stand in for the PM at Prime Minister's Questions.

In the letter, he told Mr Sunak he would "thoroughly rebut and refute" the claims against him and he was "confident that I have behaved professionally throughout".

Mr Sunak, who was absent from PMQs for the G20 summit in Bali, replied: "I know that you will be keen to address the complaints made against you and agree that proceeding in this way is the right course of action."