Characteristic Lie subring not implies ideal

ANALOGY BREAKDOWN: This is the breakdown of the analogue in Lie rings of a fact encountered in group. The old fact is: characteristic implies normal.
View other analogue breakdowns of characteristic implies normal|View other analogue breakdowns from group to Lie ring

Statement

A characteristic subring of a Lie ring need not be an ideal of the Lie ring.

Related facts

Similar facts

• Characteristic not implies derivation-invariant
• Derivation-invariant not implies characteristic

Opposite facts

• Characteristic subring implies ideal in Lazard Lie ring

Analogues in other algebraic structures

• Characteristic implies normal (in groups)
• Characteristic not implies normal in loops

Proof

Example of the simple Witt algebra

The simple Witt algebra has a characteristic subalgebra (which is hence also a characteristic subring) that is not an ideal (either in the algebra or the $\mathbb {Z}$-ring sense). More details to be inserted; for now, see here.