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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeSep 2nd 2024
   • (edited Sep 2nd 2024)
   • PermaLink
   Author: Urs
   Format: MarkdownItexhave added some further references <a href="https://ncatlab.org/nlab/revision/diff/nilpotent+Lie+algebra/8">diff</a>, <a href="https://ncatlab.org/nlab/revision/nilpotent+Lie+algebra/8">v8</a>, <a href="https://ncatlab.org/nlab/show/nilpotent+Lie+algebra">current</a>

   have added some further references

   diff, v8, current

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeSep 3rd 2024
   • PermaLink
   Author: zskoda
   Format: MarkdownItexI am sorry, but I consider the idea section both misleading/confusing maybe even wrong. The idea section, while somewhat ambiguous, in my reading refers to the notion of ad-nilpotent Lie algebra (elementwise notion), or maybe, stretching a bit, locally nilpotent Lie algebra (every finite dimensional Lie subalgebra is nilpotent), rather than a nilpotent Lie algebra (global notion). Every _finite dimensional_ ad-nilpotent Lie algebra is nilpotent, this is the Engel's theorem but in general being nilpotent is a stronger notion.

   I am sorry, but I consider the idea section both misleading/confusing maybe even wrong.

   The idea section, while somewhat ambiguous, in my reading refers to the notion of ad-nilpotent Lie algebra (elementwise notion), or maybe, stretching a bit, locally nilpotent Lie algebra (every finite dimensional Lie subalgebra is nilpotent), rather than a nilpotent Lie algebra (global notion). Every finite dimensional ad-nilpotent Lie algebra is nilpotent, this is the Engel’s theorem but in general being nilpotent is a stronger notion.

3. • CommentRowNumber3.
   • CommentAuthorzskoda
   • CommentTimeSep 3rd 2024
   • PermaLink
   Author: zskoda
   Format: MarkdownItex> A [[Lie algebra]] is **nilpotent** if repeatedly [[adjoint action|acting]] via the [[Lie bracket]] on any one of its elements with other elements eventually yields [[zero]]. This can be rectified in fact, by just requiring a **uniform** $n$ such that $ad x_1 ad x_2 \ldots ad x_n = 0$.

   A Lie algebra is nilpotent if repeatedly acting via the Lie bracket on any one of its elements with other elements eventually yields zero.

   This can be rectified in fact, by just requiring a uniform $n$ such that $ad x_1 ad x_2 \ldots ad x_n = 0$.

4. • CommentRowNumber4.
   • CommentAuthorzskoda
   • CommentTimeSep 3rd 2024
   • PermaLink
   Author: zskoda
   Format: MarkdownItexSorted out the ad-nilpotence versus nilpotence. <a href="https://ncatlab.org/nlab/revision/diff/nilpotent+Lie+algebra/11">diff</a>, <a href="https://ncatlab.org/nlab/revision/nilpotent+Lie+algebra/11">v11</a>, <a href="https://ncatlab.org/nlab/show/nilpotent+Lie+algebra">current</a>

   Sorted out the ad-nilpotence versus nilpotence.

   diff, v11, current