The (local) lifting property is characterized by lifting *-homomorphisms; it is preserved under free products amalgamated over finite-dimensional subalgebras, Exel's soft tori have the LP, and for certain algebras LLP is equivalent to Ext being a group.
Blackadar, Shape theory for C*-algebras, Math
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On the (Local) Lifting Property
The (local) lifting property is characterized by lifting *-homomorphisms; it is preserved under free products amalgamated over finite-dimensional subalgebras, Exel's soft tori have the LP, and for certain algebras LLP is equivalent to Ext being a group.