The man held over the attack on Labour MP Jo Cox is "not a violent man and is not that political", his brother has said.
Tommy Mair, 52, was detained by police near his home in Birstall, West Yorkshire.
Officers were seen searching his home not far from the scene of the attack, during which the alleged assailant reportedly shouted "Britain first".
Scott Mair, 50, told reporters on Thursday that his brother had a "history of mental illness, but he has had help".
And he told the Sun: "We are struggling to believe what has happened. My brother is not a violent man and is not that political.
"We don't even know who he votes for. I am visibly shaken at this news. I am so sorry for the MP and her family."
Mair's half-brother, Duane St Louis, 41, told ITV he believed his brother "wouldn't hurt a fly".
He told the Sun: "He's never expressed any views about Britain, or politics or racist tendencies. I'm mixed race and I'm his half-brother, we got on well."
He was speaking after it emerged that a Thomas Mair, from Batley, was named in a 2006 edition of the Springbok Cyber Newsletter, which is produced by the hard-right Springbok Club, an organisation which has called for a return to apartheid-style government in South Africa.
The online article said he was "one of the earliest subscribers and supporters of 'SA Patriot'" and was hoping to trace his whereabouts.
In 2010 a Thomas Mair, then aged 46, was quoted in his local paper talking about his mental health issues.
He volunteered at Oakwell Hall country park in Birstall in 2010 after being a patient of the Mirfield-based Pathways Day Centre for adults with mental illness, according to a Huddersfield Examiner report at the time.
He told the paper: "I can honestly say it has done me more good than all the psychotherapy and medication in the world.
"Many people who suffer from mental illness are socially isolated and disconnected from society, feelings of worthlessness are also common, mainly caused by long-term unemployment."
Pathways is described on the South West Yorkshire Partnership (SWYP) NHS Foundation Trust website as a "team that offers support to people experiencing mental health difficulties".
Rob Webster, chief executive at SWYP NHS Trust, said: "Our thoughts are with Jo's family, friends and colleagues during this tragic time.
"This is an ongoing police investigation and we are unable to comment further."
Speaking on Thursday, neighbours said Mair had lived there for more than 30 years - on his own for the last two decades since the death of his grandmother.
They did not think he had a job or drove a car, and described how he would do gardening chores for local people.
David Pickles said: "He's lived there longer than me and I've lived here since 1975. I still can't believe it. He's the last guy I would have thought of.
"He's just quiet. He kept himself to himself. He lived by himself. He's been on his own for about 20 years.
"I've never seen a lot of people visiting or anything like that, but he likes gardening. He did a lot of people's gardens round here. But he did it quietly."
Pickles said he did not know of any connection between his neighbour and Cox.
Another neighbour, who did not want to be named, said: "We've known him quite a lot of years. I've known him since I was young. He lived with his grandmother in that house, she died a few years ago. He was a bit of a loner. It's a bit of a shock - he did everyone's gardens.
"I can't say a wrong word about him, he was so quiet. It's come as a shock to everybody."