The input should be a list of Lie elements in a free Lie algebra $F$ and this list consists of the differentials of the generators, where $0_F$ is used for the zero element. The option LastWeightHomological for $F$ must have the value true. The program adds relations to the Lie algebra to get the square of the differential to be 0. Use ideal(LieAlgebra) to get these non-normalized relations or describe(LieAlgebra) to just look at them.
i1 : F1=lieAlgebra({a,b,c,r3,r4,r42},
Weights => {{1,0},{1,0},{2,0},{3,1},{4,1},{4,2}},
Signs=>{0,0,0,1,1,0},LastWeightHomological=>true)
o1 = F1
o1 : LieAlgebra
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i2 : D1=differentialLieAlgebra{0_F1,0_F1,0_F1,a c,a a c,r4 - a r3}
o2 = D1
o2 : LieAlgebra
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i3 : ideal D1
o3 = {(a a c) - (a a c)}
o3 : List
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i4 : F2=lieAlgebra({a,b,c2,c3,c5},Signs=>{0,0,1,0,1},
Weights=>{{1,0},{1,0},{2,1},{3,2},{5,3}},
LastWeightHomological=>true)
o4 = F2
o4 : LieAlgebra
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i5 : D2=differentialLieAlgebra{0_F2,0_F2,a b,a c2,a b c3}
o5 = D2
o5 : LieAlgebra
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i6 : describe D2
o6 = generators => {a, b, c2, c3, c5}
Weights => {{1, 0}, {1, 0}, {2, 1}, {3, 2}, {5, 3}}
Signs => {0, 0, 1, 0, 1}
ideal => { - (a b a), (a b a c2)}
ambient => F2
diff => {0, 0, - (b a), (a c2), (a b c3)}
Field => QQ
computedDegree => 0
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The object differentialLieAlgebra is a method function.