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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 Engelâ€™s theorem but in general being nilpotent is a stronger notion.
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 $ad x_1 ad x_2 \ldots ad x_n = 0$.
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