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It is well known that the nilradical of a finite-dimensional Lie algebra over a field of characteristic p > 0 need not be characteristic (that is, invariant under all derivations of the algebra), but is there an example of a solvable Lie algebra with non-characteristic nilradical? I've asked this question on Mathoverflow but have had no answer in nearly a month. Any ideas as to how to construct such an example would be welcome.

David Towers
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    Did you test all low-dimensional solvable Lie algebras in characteristic $p$, classified by Willem de Graaf here, for small $p$, e.g., for $p=2$ or $p=3$ ? – Dietrich Burde Mar 03 '16 at 14:48
  • MathOF link: http://mathoverflow.net/questions/230861/ – YCor Mar 03 '16 at 15:30
  • Thank you Dietrich. 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 as there is quite a lot of work involved in finding the derivations. Some can be ruled out because N(L)=L^2. Also, N(L) can't be abelian if p>2. – David Towers Mar 03 '16 at 16:13
  • I see. The computation of the derivations, though, is not much work with a CAS. – Dietrich Burde Mar 03 '16 at 18:44
  • Thank you Dietrich. I have found a 4-dimensional example over GF(2). – David Towers Mar 07 '16 at 17:38

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