MathOverflow Asked by David Towers on November 7, 2021
Say that an ideal in a Lie algebra is characteristic if it is invariant under every derivation of the algebra.
It is well known that the nilradical of a finite-dimensional Lie algebra over a field of characteristic $p > 0$ need not be characteristic, but is there an example of a solvable finite-dimensional Lie algebra with non-characteristic nilradical?
It is clear that the nilradical is invariant under automorphisms (this is also sometimes used a a definition of characteristic ideal). For an ideal in a finite-dimensional Lie algebra in characteristic zero, the latter is equivalent to being characteristic. But this is not the case in positive characteristic.
The question was somewhat answered by the OP in comments of the MathSE duplicate, so let me copy the relevant parts here for the record:
Did you test all low-dimensional solvable Lie algebras in characteristic ?, classified by Willem de Graaf here, for small ?, e.g., for ?=2 or ?=3 ? – Dietrich Burde Mar 3 '16
I've had a look at these, but there are none amongst the 3-dimensional algebras. I haven't yet examined all of the 4-dimensional ones (...). – David Towers Mar 3 '16
I have found a 4-dimensional example over $mathrm{GF}(2)$. – David Towers Mar 7 '16
Answered by YCor on November 7, 2021
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