By V. A. Krechmar

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I) If β is a limit ordinal, set G[β] = α<β G[α] . 36 Panoramic Overview (ii) If β = α + 1, set G[β] = [G, G[α] ]. For cardinality reasons, there is a smallest ordinal δ such that G[δ+1] = G[δ] . Set = G[δ] . Then G is said to be transfinitely nilpotent if G[∞] = {0}. If G is transfinitely nilpotent and δ ≤ ω, then G is called countably nilpotent. If δ is finite and G[δ] = {0}, then G is called nilpotent. G[∞] Since G(α) ⊆ G[α] , any transfinitely nilpotent Lie group is transfinitely solvable. As we are dealing with topological groups, we have topological versions of these concepts as well.

G is countably topologically nilpotent. g is pronilpotent. For every pair x, y of elements in g the vector space endomorphism ad x satisfies limn (ad x)n y = 0. A comparison of the preceding two results produces a certain difference in the structure of condition (vi) in the two cases. This in indicative of the fact, that the treatment of the two cases is not entirely parallel. In the case of nilpotency we shall show that a pro-Lie algebra g has a unique largest pronilpotent ideal which is called its nilradical and is denoted by n(g).

9). Let G be a subgroup of a topological group. We set G[[0]] = G and define sequences of closed normal subgroups G[[α]] indexed by the ordinals α, card α ≤ card g via transfinite induction. Assume that G[[α]] is defined for α < β. (i) If β is a limit ordinal, set G[[β]] = α<β G[[α]] . (ii) If β = α + 1, set G[[β]] = [G, G[[α]] ]. For cardinality reasons, there is a smallest ordinal δ such that G[[δ+1]] = G[[δ]] . Set G[[∞]] = G[[δ]] . Then G is said to be transfinitely topologically nilpotent, if G[∞] = {0}.