The man being held in connection with
has been named as Thomas Mair, who was described as a "loner" with a history of mental health problems who had previously subscribed to a far-right magazine.
Mair, 52, who was arrested by armed officers shortly after the attack, had spoken about receiving "psychotherapy and medication", and was described by his younger brother as having "a history of mental illness".
Despite being born in Kilmarnock, Scotland, a decade-old website posting identified Mair as a subscriber to S. A. Patriot, a South African magazine that was published by the pro-apartheid group, the White Rhino Club.
The club describes the magazine's editorial stance as being against "multi-cultural societies" and "expansionist Islam". A blog post attributed to the group, dated January 2006, described Mair as "one of the earliest subscribers and supporters of S. A. Patriot."
Mair, whose home is a small, semi-detached house on the Field Head council estate, in Birstall, is said to have lived in the property for 40 years.
Neighbours said that he lived with his grandmother, until she died twenty years ago, and had never had any full-time employment.
His brother, Scott, 49, said he had "a history of mental illness". Scott Mair told reporters he had wept when he heard about the killing of Mrs Cox.
He said: "I am struggling to believe what has happened. My brother is not violent and is not all that political. I don't even know who he votes for. He has a history of mental illness, but he has had help."
Visibly shaking, he added: "I cried when I heard. I am so sorry for her and her family."
In 2011, Mair spoke of how he had volunteered to work as a groundsman at the nearby Oakwell Hall County Park, which had helped ease his mental health problems.
He told a local newspaper: "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.
"All these problems are alleviated by doing voluntary work. Getting out of the house and meeting new people is a good thing, but more important in my view is doing physically demanding and useful labour.
"When you have finished there is a feeling of achievement which is emotionally rewarding and psychologically fulfilling.
"For people for whom full-time, paid employment is not possible for a variety of reasons, voluntary work offers a socially positive and therapeutic alternative."
Last night his house was cordoned off and under police guard as neighbours spoke of the "very quiet but very helpful" suspect.
Kathleen Cooke, 62, said: "I am really shocked. He walked past my house this morning and said hello like he always does. He was wearing a grey T shirt and his white baseball cap like he always does and he was carrying a small rucksack.
"He is just a quiet bloke who keeps himself to himself. "He is very helpful and he helps local people with their gardens. There is one neighbour who is a bit frail and he keeps her garden tidy. He has helped me cut my hedge a couple of times.
"He has lived here for 40 years and has never been in any trouble and has never caused any trouble. He sometimes used to shout at the local kids if they played too near his house but that is fairly normal.
"I don't think he belonged to any political party and I never heard him express any views about Europe or anything like that. To us, he was just Tommy, a local bloke we all knew.
"He did not have a job but I think he went down to the Job Centre to help people on the computer from time to time - I think it was something he had to do to get his dole money."
David Pickles, 62, an assembly worker , said: "I was speechless when I heard the news - I could not believe it. He is just a bit of a loner who keeps himself to himself but always says hello. He seemed to like his own company but I would not say he was unfriendly - he would always pass the time of day.
"I know he spent a lot of time in the library in town and liked to go on the computers there - I don't know what he was looking up."