  The Local Lifting Property (LLP) is a localized version of projectivity for
completely positive maps between $\mathrm{C}^*$-algebras. Outside of the
nuclear case, very few $\mathrm{C}^*$-algebras are known to have the LLP. In
this article, we show that the LLP holds for the algebraic contraction
$\mathrm{C}^*$-algebras introduced by Hadwin and further studied by Loring and
Shulman. We also show that the universal Pythagorean $\mathrm{C}^*$-algebras
introduced by Brothier and Jones have the Lifting Property.
