The parents of six children murdered their two teenage sons the day after a bid to poison them failed.
Sarah Barrass and Brandon Machin, who is her half-brother, strangled Tristan and Blake Barrass, aged 13 and 14, in Shiregreen, Sheffield, in May.
The court heard how Barrass, 35, would regularly tell her children: "I gave you life, I can take it away."
Barrass and Machin, 39, were both sentenced to a minimum of 35 years in prison at Sheffield Crown Court.
They have both previously admitted murder, conspiracy to murder all six of their children, including Blake and Tristan, and five counts of attempted murder.
The court heard how Barrass strangled Tristan with her dressing gown cord, before Machin strangled Blake with his hands.
They then put plastic bags over the boys' heads, suffocating them.
South Yorkshire Police said Barrass, of Gregg House Road, Shiregreen, and Machin, of Burngreave Road, had been in a secret sexual relationship for years.
Fearing they would be found out by the authorities and their children taken into care, they hatched a plot to kill them. Police said the plan was for Machin to discover what had happened and raise the alarm.
On the evening of 23 May, Barrass tried to poison the four eldest children, by collecting tablets prescribed to one of the children for ADHD and forcing them to swallow them.
Kama Melly QC, prosecuting, said: "None of the children wanted to take the tablets but were forced to do so.
"The defendants expected the tablets to kill the children overnight."
When it became apparent the plan had failed, Barrass began to search online for other ways of killing her children, including suffocation, strangulation and drowning.
She contacted Machin to tell him they were still alive and the pair then strangled the boys and placed bin bags over their heads "to ensure their certain death", Ms Melly said.
The defendants then ran a bath and repeatedly tried to drown one of the younger children.
When that too failed, Barrass took the surviving children - two of whom are under the age of 13, and two under three - to the bedroom and phoned the police.
The court heard Barrass had previously approached the local authority to ask for help with her children.
Ms Melly said the mother sent a message to a friend which said: "I've thought of every possible solution to this mess.
"I love my kids too much to kill them, I can't put them into care for the same reason."
Bryan Cox QC, mitigating for Barrass, said she was "profoundly damaged by her childhood".
He said: "The defendant was desperate to prevent her children being taken into care.
"She couldn't cope with the prospect of them being removed."
The court heard she told police she planned to kill the younger two children and herself, after the older four had died.
Mr Justice Goss, sentencing, said to Barrass: "You considered your love for them and fear of being parted from them entitled you to take their lives as well as your own."
Speaking after the sentencing, Edmund Hulbert from the Crown Prosecution Service said: "This was an appalling crime in which two young lives were lost, and a family torn apart, leaving a community in shock.
"Two of the surviving children witnessed their older siblings being attacked and the trauma that all the children have experienced, and will continue to experience, is unimaginable.
"It is paramount now that the surviving children are allowed to rebuild their lives in peace."
Matthew Saunders, a friend of the murdered boys, said outside court: "A piece of all our hearts died on 24 May 2019, which we will never get back.
"Blake and Tristan leave a huge empty void in our lives, and we did not get chance to say goodbye.
"We are relieved justice has been served, but it should never have come to this."