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Suppose $(g; k ,p)$ is a reductive symmetric Lie algebra. i.e. $k$ is a sub-algebra of $g$, $[k,p] \subset p$ , $[p,p] \subset k$ and $g= k \oplus p$.

This is actually from Lepowsky and McCollum's paper on Cartan Subspaces of Symmetric Lie Algebras (Transactions of the American Mathematical Society Vol. 216 (Feb., 1976), pp. 217-228), theorem 5.1.

"... Since the Killing form $B$ of $g$ is a non-singular trace form on $k$, then $k$ is reductive."

My questions:

  1. What does non-singular trace form mean?

  2. Why is $k$ reductive?

Torsten Schoeneberg
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hhpolis
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1.) A nonsingular trace form on a Lie algebra $L$ over a field $K$ is a bilinear form $f\colon L\times L\rightarrow K$ given by $f(x,y)=tr(\rho(x)\rho(y))$, which is non-degenerate as a bilinear form; here $\rho\colon L\rightarrow \mathfrak{gl}(V)$ is a Lie algebra representation. For example, with $\rho=ad$ we have the Killing form. It is nonsingular iff the Lie algebra is semisimple, over characteristic zero.

2.) If you read the proof of Theorem 5.1 in the paper, it says "Since the Killing form $B$ of $g$ is a nonsingular trace form on $k$, $k$ is reductive." And then they even give two references for this well-known result: "see either [5, §2.9] or [1, 1.7.3] ". You can also find this in other common books on Lie algebras.

Dietrich Burde
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    +1. For posterity's sake, reference 1 is Dixmier's Algèbres enveloppantes, Gauthier-Villars, Paris, 1974, and in reference 5 the authors Lepowsky and McCollum quote their own book Elementary Lie algebra theory, Yale University Lecture Note Series, 1974. Another reference for this is Bourbaki's Lie Groups and Algebras, ch. I paragraph 6 no. 4 proposition 5, which gives equivalent conditions for a Lie algebra to be reductive, among them: That is has a finite-dimensional representation whose associated bilinear trace form is non-degenerate. – Torsten Schoeneberg Mar 01 '23 at 18:57
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    @TorstenSchoeneberg Thank you for the references! I looked it up in Bourbaki's Lie Groups and Lie Algebras, Chapter 1-3, Springer Verlag, English Edition 1989. It is also Proposition 5 is on page 56 there, and it is condition (d). This book is an excellent reference work. Although many results are contained in exercises (for example page 73-110). – Dietrich Burde Mar 01 '23 at 19:26