The generators in the $i$th set (beginning with $i=0$) in the inputs of holonomy generate a subalgebra of the holonomy Lie algebra $H$, and the output of holonomyLocal(i,H) is this Lie subalgebra. If the set is of size $k$, then the local Lie algebra is free on $k$ generators if the set belongs to the first input set, and it is free on $k-1$ generators in degrees $\ge 2$ if it belongs to the second input set.
i1 : H=holonomy({{a1,a2},{a3,a4}},{{a1,a3,a5},{a2,a4,a5}})
o1 = H
o1 : LieAlgebra
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i2 : describe holonomyLocal(1,H)
o2 = generators => {a3, a4}
Weights => {{1, 0}, {1, 0}}
Signs => {0, 0}
ideal => {}
ambient => LieAlgebra{...10...}
diff => {}
Field => QQ
computedDegree => 0
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i3 : describe holonomyLocal(2,H)
o3 = generators => {a1, a3, a5}
Weights => {{1, 0}, {1, 0}, {1, 0}}
Signs => {0, 0, 0}
ideal => {(a3 a1) - (a5 a3), (a5 a1) + (a5 a3)}
ambient => LieAlgebra{...10...}
diff => {}
Field => QQ
computedDegree => 0
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The object holonomyLocal is a method function.