The vector space $D$ of graded derivations from $L$ to $L$ with the identity map as defining map, see LieDerivation, is a graded Lie algebra. If $L$ has a differential $del$, then $D$ is a differential graded Lie algebra with differential $d$\ \to\ [$del$,$d$].
i1 : L = lieAlgebra{a,b}/{a a a b,b b b a}
o1 = L
o1 : LieAlgebra
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i2 : d0 = lieDerivation{a,b}
o2 = d0
o2 : LieDerivation
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i3 : d2 = lieDerivation{a b a,0_L}
warning: the derivation might not be well defined, use isWellDefined
o3 = d2
o3 : LieDerivation
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i4 : d4 = lieDerivation{a b a b a,0_L}
warning: the derivation might not be well defined, use isWellDefined
o4 = d4
o4 : LieDerivation
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i5 : describe d2 d4
o5 = a => (a b a b a b a)
b => 0
map => id_L
sign => 0
weight => {6, 0}
source => L
target => L
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i6 : describe d0 d4
o6 = a => 4 (a b a b a)
b => 0
map => id_L
sign => 0
weight => {4, 0}
source => L
target => L
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