Lacey has updated Battle of Brothers: The Inside Story of a Family in Tumult with added chapters and new material, and in new excerpts in The Times newspaper in the U.K. published on Saturday, the veteran writer outlines more details of their split.
William, 38, and Harry, 36, were originally said to have fallen out over William's doubts about the speed at which his brother was moving with his then-girlfriend Meghan Markle.
Lacey writes that although relations between the princes had eased by the time of Harry and Meghan's May 2018 wedding, they worsened again in October 2018 when William learned that a senior palace aide was making allegations of bullying by Meghan to staffers.
Gareth Fuller/WPA Pool/Getty Images Prince William and Prince Harry
When William heard those claims and confronted his brother, they had a "fierce and bitter" argument, Lacey writes - leading William to start moves to remove Harry and Meghan from the joint office, or household, they shared.
"William threw Harry out," a friend tells Lacey.
Max Mumby/Indigo/Getty Images Prince Harry and Prince William
When the bullying claims first emerged, in the The Times in the U.K. in March, the Duchess of Sussex's office strongly refuted the allegations. The Palace set up an independent inquiry into the claims and how they were handled.
Lacey's book describes an angry confrontation between the princes - first over the phone and then in person when William asked Harry about the bullying claims.
"When Harry flared up in furious defense of his wife, the elder brother persisted," reads an excerpt. "Harry shut off his phone angrily, so William went to speak to him personally. The prince was horrified by what he had just been told about Meghan's alleged behavior, and he wanted to hear what Harry had to say. The showdown between the brothers was fierce and bitter."
Representatives for Kensington Palace, where William has his office, did not comment on the claims made in the book when contacted by PEOPLE. A representative for the Sussexes also declined to comment.
Phil Harris - WPA Pool/Getty Prince William, Kate Middleton, Prince Harry and Meghan Markle
Lacey also claims that Kate Middleton had been "wary of [Meghan] from the start" and says that William told a friend he thought she had an "agenda," and that the allegations he was now hearing in fall of 2018 appeared to support his initial reservations about Meghan.
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Lacey quotes a Kensington Palace staffer as saying that "people felt run over" by Meghan and "didn't know how to handle" her.
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The new excerpts come as the brothers prepare to reunite to unveil a statue commemorating the life of their late mother, Princess Diana, on July 1, which would have been her 60th birthday.
In the first edition of the book, Lacey explored the rift, rooting the schism in their parents' turbulent marriage.
"Both brothers have been damaged by their upbringings; both have reacted by finding different solutions," says the author, who is also the historical consultant for Netflix's The Crown. "There is so much pain and trauma in this story, going right back to the beginning."
He said that mounting tensions accompanied Harry and Meghan Markle's whirlwind relationship.
"The fundamental conflict was between the two males who had known each other all their lives and had never hesitated to tell each other exactly what they thought and felt," Lacey wrote. "William worried that his brother was moving too fast in his courtship - and he did not shrink from saying as much when Harry started talking about getting hitched to Meghan quite soon."