Alesbrook, previously of Upper Greenhill Gardens, Matlock in Derbyshire, met his ex-partner Shemwell in November 2021.
He previously denied murdering Elijah but was unanimously convicted after a five-week trial in July.
The court heard Alesbrook caused whiplash-type injuries, bleeding on the brain, and multiple bone fractures to Elijah.
Video taken by Shemwell of her son unresponsive with a floppy arm on 1 January was shown to the jury.
Another video showed him “gasping for breath” before Shemwell phoned emergency services on the evening of 2 January, and he was taken by an ambulance to Queen’s Medical Centre in Nottingham.
Elijah died three days later on 5 January with fatal head injuries that were described in court as “catastrophic”.
Shemwell, who was 21 at the time but is now 24, was sentenced to three years’ imprisonment at the same court after she admitted two counts of child cruelty in December last year.
