The England international midfielder, who Saints plucked from Cherries' centre of excellence at the age of 12 in September 2000, was unveiled by the Reds having completed his switch for an undisclosed fee.
It has been widely reported that £25million was paid for the 26-year-old but the Daily Echo understands that Cherries, entitled to 25 per cent of the proceeds, received in the region of £4million from the deal.
Blake said that the Dean Court club were bound by confidentiality over their share of the cash but that the protracted negotiations had required flexibility from everyone concerned.
Blake told the club website: "The transfer fee was undisclosed which means our cut will also remain undisclosed, but it was certainly not as simple as 25 per cent of any fee which has been reported by the press over the past few weeks.
"There were lots of small sacrifices that had to be made by all parties for the deal to happen, but I think each club involved, and the player himself, can be very happy with the final outcome."
Cherries initially received £2,000 for the two years Lallana was registered with the club, together with a £1,000 finding fee.
Further stage payments were made when he signed scholarship (£5,000) and professional forms (£10,000) for Saints, while an FA tribunal also made provision for the Dorset club to receive the 25 per cent sell-on.
A firm fans' favourite at St Mary's, Lallana shot to national prominence during Southampton's two seasons back in the Premier League and earned his first England call-up in November last year before being named on the six-man shortlist for the PFA Player of the Year award.
Speculation over his switch to Anfield was rife prior to jetting out for the World Cup in Brazil, with the anticipated deal eventually concluded following a drawn-out process.
Meanwhile, Blake paid tribute to Cherries staff past and present who played their part in Lallana's progress.
Blake added: "I am delighted for the club, and those that supported Adam during his time here. I know it is always disappointing for supporters to lose such a great talent at such a young age but we have to give credit to those who dealt with his move to Southampton.
"The development of young talent has always been a massive part of the make-up at AFC Bournemouth and, although Adam left at an earlier age than others, we should still view him as success story for our academy.
"All of the players who have gone on to achieve great things are now used by our current academy coaches as a fine example of what can be achieved with enough hard work and dedication."