Carlyle acquired Nuneaton-based Holland & Barrett as part of its $3.8bn (£3bn) purchase in 2010 of US firm Nature's Bounty, now NBTY.
The chain, which has more than 1,300 stores worldwide, is expected to change hands in September.
Holland & Barrett was founded by William Holland and Alfred Barrett in Bishop's Stortford, Hertfordshire, in 1870.
They initially sold groceries and clothing, but later split the two into separate businesses. The grocery business was sold to Alfred Button & Sons in the 1920s, but the original name was retained.
The company eventually started focusing on health foods and changed hands several times. It now employs more than 4,000 people.
"Holland & Barrett is a clear market leader in the UK health and wellness retail market, with attractive growth positions in other European and international markets," said L1 Retail managing partner Stephan DuCharme.
"We believe that the company is well positioned to benefit from structural growth in the growing £10bn health and wellness market and has multiple levers for long-term growth and value creation."
The purchase is the first by L1 Retail, which was set up in late 2016.
It aims to invest $3bn in a small number of retail businesses that it believes can be market leaders by "moving with and leading long-term trends".
The fund's advisory board includes John Walden, the former chief executive of Home Retail Group.
Other members are Karl-Heinz Holland, who was chief executive of Lidl Group, the German supermarket chain, and Clive Humby, one of the founders of dunnhumby, which came up with the idea for Tesco's Clubcard.
L1 also has funds focused on energy, technology and health.
Peter Aldis, Holland & Barrett chief executive, is to stay on. He said: "We are delighted to now be in partnership with the L1 Retail team and its advisory board of internationally-renowned retailers."
Mr Fridman is best known for his role as chief executive of BP's Russian joint venture TNK between 2003 and 2012, when it was sold to Rosneft for $56bn.
He used the proceeds from the sale of his stake to set up L1, which also has investments in the telecoms, technology and energy sectors.
His oil and gas interests stretch from Algeria and Libya to Poland and Norway.Mervyn Davies, the former Standard Chartered chief executive who is now Lord Davies of Abersoch, is chairman of L1 Holdings.