Let \({\mathbb {F}}\) be a field of algebraically closed and L be a finite dimensional Lie algebra over field \( {\mathbb {F}}\) . \(L'=[L,L]\) denotes the derived subalgebra of L. Following the analogy with group theory, we define the subalgebra D(L) of L to be the intersection of the normalizer of the derived subalgebras of all subalgebras of L. In a Lie algebra L, this is an ideal of L, allowing the definition of the ascending series: set \(D_0(L) = 0\), \(D_{i+1}(L)/D_i (L) =D(L/D_i (L))\) for \(i \ge 1\), \(D_{\infty }(L)\) denotes the terminal term of the ascending series. It is proved that L is solvable if and only if \(L=D_{\infty }(L)\).

中文翻译：

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导出次子与有限维李代数的可解性

令\（{\ mathbb {F}} \）是代数封闭的字段，L是字段\（{\ mathbb {F}} \）上的有限维李代数。\（L'= [L，L] \）表示L的派生子代数。在与群理论的类比，我们定义了子代数d（大号的）大号是所有子代数的导出子代数的归一化的交点大号。在一个李代数大号，这是一种理想的大号，允许递增系列的定义：组\（D_0（L）= 0 \） ，\（D_ {I + 1}（L）/ D_i（L）= d（L / D_i（L））\）为\（i \ ge 1 \），\（D _ {\ infty}（L）\）表示升序的末项。证明当且仅当\（L = D _ {\ infty}（L）\）时，L是可解的。