Compact connected simple Lie skew braces are rigid (trivial on S^1 or have simple groups and trivial/almost-trivial brace); all compact connected solvable ones are trivial, but noncompact simple examples with solvable groups exist.
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On simple compact Lie skew braces
Compact connected simple Lie skew braces are rigid (trivial on S^1 or have simple groups and trivial/almost-trivial brace); all compact connected solvable ones are trivial, but noncompact simple examples with solvable groups exist.