Ben Purdy, 18, was shot in the head in Bermondsey after confronting his girlfriend's ex, Michael Bagnall, about "threatening" messages.
Bagnall, 22, of Hospital Way, Lewisham, must serve at least 28 years.
His uncle Andrew Bayne, 37, of Trundleys Terrace, Lewisham was sentenced to at least 30 years.
During the trial at the Old Bailey, jurors heard that they both suffered from learning disabilities, but Judge Christopher Moss said they were still "able to organise and carry out this dreadful murder".
As they were led from the dock, Bagnall launched into a tirade of abuse at the judge and shouted out "I'm glad he's dead" in the courtroom packed with Ben's family.
In a statement, Ben's mother, Joanne Treston-Smith, said: "The void that Ben's death has left will never be filled. We will always have him missing in our lives."
The court heard that the murder followed a feud between Michael Bagnall and the victim, who was going out with the Bagnall's ex-girlfriend.
Bagnall was said to have resented the new relationship and sent the girl "threatening messages" by phone or Facebook.
On the day of his murder, Ben Purdy decided to confront Bagnall, tracking him down in which led to a "skirmish" in Bermondsey, south London.
After that encounter, the jury heard Bagnall and and his uncle Andrew Bayne decided Mr Purdy needed to be "taught a lesson".
The pair armed themselves with a self loading pistol and an array of other weapons and got in a car to scour the streets for Mr Purdy and his friends.
When they caught up with them, Bayne shot Mr Purdy in the head on 23 November 2014. He died in hospital the next day.

Paul Hemming was jailed for beating partner Natalie Hemming, 31, to death in the lounge of their Milton Keynes home as their children slept, last May.
But their son woke up and after seeing his mother's body wrapped in a rug, showed police where it happened.
The family has been speaking ahead of a documentary about the murder case.
The body of Ms Hemming - who shared a surname with her partner but they were not married - was found in Hertfordshire woodland three weeks after she disappeared.
More news from Buckinghamshire
She was reported missing two days after she vanished by her mother, who voiced concerns to police about Hemming.
Hemming, 43, who had murdered her after hearing she was having an affair, was jailed for at least 20 years in November.
Ms Hemming's sisters Kerry Hammond and Joanne Beverley have told the BBC Hemming was a "very charming, smiley, happy guy that could quite frankly sell ice to Eskimos".
However, he was a manipulative and controlling man with a violent temper who could "go off the handle". She once needed hospital treatment after he threw a phone at her head.
The violence culminated in the killing of Ms Hemming at the hands of her partner, some of which was witnessed by the couple's son.
"He's sitting on that chair swinging his legs and he just blurts out this whole little story about unfortunately seeing mummy wrapped in a rug," his aunt Ms Beverley said.
"He drew this map of the living room and he showed them where he'd seen Paul cleaning the floor and actually that is exactly where they found the evidence that they needed to be able to charge him."
A year after the murder, the child was finally coming to terms with what his father had done, Ms Hammond said.
The story of how Hemming became the subject of a murder inquiry before his victim's body was found, is told in a Channel 4 documentary Catching a Killer, to be shown at 21:00 BST.
A man and his uncle have received life sentences after being found guilty of shooting dead a teenager in south London to "teach him a lesson".
[NEXT_CONCEPT]
A six-year-old boy whose evidence helped convict his father of his mother's murder "finally believes Daddy did do it," his family has said.