nLab Jacobson's lemma

Statement

Proposition

(Jacobson lemma) If $A$ and $B$ are matrices over a field of characteristic zero and $A$ commutes with $A B - B A$ then $A B - B A$ is nilpotent.

References

Due to:

• Nathan Jacobson, p. 877 in: Rational methods in the theory of Lie algebras, Ann. of Math. 36 (1935) 875-881 [doi:10.2307/1968593, jstor:1968593]

Further discussion:

• Irving Kaplansky: Jacobson’s Lemma revisited, Journal of Algebra 62 2 (1980) 473-476 [doi:10.1016/0021-8693(80)90196-9]