In her sentencing remarks, she said Mr Seeshai had "everything to live for" and described his killing as "horrific and shocking".
She described BGI as "extremely vulnerable" and said he had admitted buying the murder weapon from a friend for £40 about a month before the attack.
She continued he had been "groomed and exploited" by others and had "very many adverse childhood experiences".
The second defendant, CMB, had a supportive and loving relationship with his parents and was not previously known to police.
Explaining her reasons for the length of the minimum terms, the judge said mitigating factors included the fact the attack was not premeditated, and the defendants' age.
However, in written submissions to Thursday's hearing, the Solicitor General told the Court of Appeal the sentences were "significantly too short".
Paul Jarvis, representing the Solicitor General in court, said the judge "made significantly too much allowance" for mitigation.
Both boys attended the hearing by videolink from separate facilities.