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An extended derivation (endomorphism) of a (restricted) Lie algebra $L$is an assignment of a derivation (respectively) of $L^{\prime }$for any (restricted) Lie morphism $f:L\to <\#comment/>L^{\prime }$, functorial in $f$in the obvious sense. We show that (a) the only extended endomorphisms of a restricted Lie algebra are the two obvious ones, assigning either the identity or the zero map of $L^{\prime }$to every $f$; and (b) if $L$is a Lie algebra in characteristic zero or a restricted Lie algebra in positive characteristic, then $L$is in canonical bijection with its space of extended derivations (so the latter are all, in a sense, inner). These results answer a number of questions of G. Bergman. In a similar vein, we show that the individual components of an extended endomorphism of a compact connected group are either all trivial or all inner automorphisms. 

We prove a number of results on the survival of the type-I property under extensions of locally compact groups: (a) that given a closed normal embedding $\mathbb{N}⊴<\#comment/>\mathbb{E}$of locally compact groups and a twisted action $(\mathrm{\alpha <\#comment/>},\mathrm{\tau <\#comment/>})$thereof on a (post)liminal $C^{\ast <\#comment/>}$-algebra $A$the twisted crossed product $A{⋊<\#comment/>}_{\mathrm{\alpha <\#comment/>},\mathrm{\tau <\#comment/>}}\mathbb{E}$is again (post)liminal and (b) a number of converses to the effect that under various conditions a normal, closed, cocompact subgroup $\mathbb{N}⊴<\#comment/>\mathbb{E}$is type-I as soon as $\mathbb{E}$is. This happens for instance if $\mathbb{N}$is discrete and $\mathbb{E}$is Lie, or if $\mathbb{N}$is finitely-generated discrete (with no further restrictions except cocompactness). Examples show that there is not much scope for dropping these conditions. In the same spirit, call a locally compact group $\mathbb{G}$type-I-preserving if all semidirect products $\mathbb{N}⋊<\#comment/>\mathbb{G}$are type-I as soon as $\mathbb{N}$is, andlinearlytype-I-preserving if the same conclusion holds for semidirect products $V⋊<\#comment/>\mathbb{G}$arising from finite-dimensional $\mathbb{G}$-representations. We characterize the (linearly) type-I-preserving groups that are (1) discrete-by-compact-Lie, (2) nilpotent, or (3) solvable Lie. 

We prove a number of results having to do with equipping type-I\mathrm{C}^*-algebras with compact quantum group structures, the two main ones being that such a compact quantum group is necessarily co-amenable, and that if the\mathrm{C}^*-algebra in question is an extension of a non-zero finite direct sum of elementary\mathrm{C}^*-algebras by a commutative unital\mathrm{C}^*-algebra then it must be finite-dimensional.