Blake and Tristan Barrass were strangled by their parents at their home in Gregg House Road, Shiregreen, in May.
Their mum Sarah Barrass, 35, who had an incestuous relationship with her half-brother Brandon Machin, 39 – the father of the children – had feared the youngsters would be taken into care.
She plotted to kill them all and then herself but her plan failed.
The incestuous couple, who admitted murder, attempted murder and conspiracy to murder, were jailed for life and ordered to serve a minimum of 35 years behind bars yesterday.
Speaking outside Sheffield Crown Court after they were sentenced, family friend Matthew Saunders said: "Blake and Tristan were two beautiful, confident, and outgoing boys, who both had a bright future ahead of them.
“They had so many friends and brightened up any room they were in, especially Tristan who loved to dye his hair bright colours.
"Daily life will never be the same. The boys have left behind younger siblings whose lives have been turned upside down. They adored their older brothers, and looked up to them.
“The boys have also left behind close family friends and younger children who looked up to them, and saw them on a regular basis. This tragedy hasn’t just affected those close but the whole community too.
"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.
"We would like to thank the emergency services who attended on 24 May 2019 and did everything they possibly could to save Blake and Tristan. We cannot thank you enough.
“We would also like to thank Family Liaison Officer DC Little for all her help and support over the last six months and for making it that little bit easier when we were struggling.
"We cannot put into words the pain and emptiness we feel, but seek some comfort knowing Blake and Tristan will always be remembered, and how loved both boys were. Thank you."