 Solution

ID: past-exam-of-the-lie-algebras-and-representation-theory-of-symmetric-groups-course-of-the-university-of-oxford/2008/a/1/solution

L_{1} = L

and

L_{n + 1} = [L, L_{n}]

, where brackets of subspaces mean their linear span. Each term is an ideal. Inductively, if

L_{n}

[z, [x, y]] = [[z, x], y] + [x, [z, y]] \in [L, L_{n}] (z, x \in L, y \in L_{n}),

L_{n + 1}

is an ideal as well. Also

L_{n + 1} \subseteq L_{n}

. Nilpotence means that this descending series eventually reaches zero; class

c

L_{c + 1} = 0

with

L_{c} \neq = 0

for nonzero

L

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