The case opened a week after the government, contending that an immediate inquiry would embolden insurgents in Iraq, survived a bid in parliament by opposition parties to force a wide-ranging inquiry into its conduct in Iraq.
"The soldiers' families are entitled to know that lessons have been learned and this won't happen in the future," said Phil Shiner, a lawyer representing the families of two soldiers killed in Iraq.
He was speaking as the Court of Appeal began the first of three days of hearings into arguments for an inquiry. The request came from lawyers representing relatives of four soldiers who died in military action between 2003 and 2005.
The families contend that Britain is implicitly required to hold an inquiry under Article 2 of the European Convention on Human Rights, which protects the "right to life."
They want the government to explain how 13 pages of equivocal advice from Attorney General Lord Peter Goldsmith on March 7, 2003, changed within 10 days to a one-page unequivocal advice that an invasion would be legal.
"We're not asking this court to say that the military orders were lawful or unlawful, or the war was lawful or unlawful," said Shiner, speaking outside the court in central London.
"What we do say is that mistakes were made. Cabinet was never shown Goldsmith's advice. He changed his mind. He didn't go to outside opinion from a barrister specialising in international law," Shiner told reporters.
In papers before the court, government lawyers argued that the case was essentially a political matter for which the government was answerable to parliament and ultimately the electorate.
They said that if the judges accepted that it was a human rights case, it would "mark a significant, constitutional shift."
Shiner rejected their argument: "I don't personally believe that the court are going to be seduced into thinking somehow the world will end if we are allowed our inquiry."
If the families win the case for an independent inquiry, Shiner said, the government could challenge that ruling in the House of Lords, the highest appeal court in Britain.
If an inquiry were to go ahead, the government would draw up its terms of reference.
A High Court judge ruled in December 2005 that the families did not have an arguable case to go to a full judicial review hearing.
However, the appeal judges gave permission for a hearing "because the case raises questions of considerable general importance which should, we think, be finally decided after full argument."
At the same time they have warned the families that they face "formidable hurdles" in trying to establish their case.
Outside court, Rose Gentle, whose 19-year-old son Gordon was killed by a roadside bomb in June 2004 in the southern Iraqi city of Basra, challenged Prime Minister Tony Blair to be "man enough" to order a full public inquiry now.
After the government last week survived moves in parliament for a probe, Blair said it had not ruled out a "retrospective inquiry" once the situation in Iraq allowed.

Copyright AFP (Agence France-Presse), 2006