Ashok Kalyanjee, 52, maintains there was a failure by psychiatrists to identify that he was suffering from a paranoid personality disorder at the time of the offences.
If he succeeds he could face standing trial for the fatal assaults on Paul Ross, 6, and his younger brother Jay, 2, with a jury left to decide if it was murder or the lesser offence of culpable homicide.
The Crown is contesting the challenge at the Appeal Court in Edinburgh and maintains there was no miscarriage of justice.
Kalyanjee, a former student who was born in India, had collected the boys who lived with their mother in Glasgow.
He drove them to a layby in the Campsie Fells, near Lennox-town and killed them on May 3 in 2008.
The children's throats were cut and he poured petrol over them and himself. He was jailed for life.