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Let g be a finite-dimensional real simple Lie algebra of compact type and let <-,-> denote the positive-definite inner product induced from the negative of the Killing form. Let \Omega denote the trilinear map defined by

       \Omega(X,Y,Z) = <[X,Y],Z> .

It is easy to see that it is alternating, because of the ad-invariance of the Killing form. Let us call a subspace S of g isotropic if \Omega vanishes identically when restricted to S; that is, if

      \Omega(X,Y,Z) = 0   whenever X,Y,Z belong to S.

In other words, S is isotropic iff [S,S] \subset S^\perp, where \perp means the perpendicular complement relative to the Killing form.

Furthermore we say that an isotropic subspace is maximal if it is not properly contained in an isotropic subspace. It is not hard to show that

            S is maximal isotropic iff [S,S] = S^\perp.

The question is how to characterise the maximal isotropic subspaces of g.

It is easy to see that the maximally isotropic subalgebras are precisely the Cartan subalgebras, but I am interested in subspaaces which are not necessarily subalgebras.

The only examples I know are those for which S = k^\perp and k < g a subalgebra, whence

              g = k \oplus S

is a symmetric decomposition corresponding to the compact riemannian symmetric space G/K.

Question: Are there any other maximal isotropic subspaces?

Let $\bf g$ be a finite-dimensional real simple Lie algebra of compact type and let $\left<-,-\right>$ denote the positive-definite inner product induced from the negative of the Killing form. Let $\Omega$ denote the trilinear map defined by $$\Omega(X,Y,Z) = \left<[X,Y],Z\right> .$$ It is easy to see that it is alternating, because of the ad-invariance of the Killing form. Let us call a subspace $S\subset{\bf g}$ isotropic if $\Omega$ vanishes identically when restricted to $S$; that is, if $$\Omega(X,Y,Z) = 0, \forall X,Y,Z \in S.$$

In other words, $S$ is isotropic iff $[S,S] \subset S^\perp$, where ${}^\perp$ means the perpendicular complement relative to the Killing form.

Furthermore we say that an isotropic subspace is maximal if it is not properly contained in an isotropic subspace. It is not hard to show that $S$ is maximal isotropic if and only if $[S,S] = S^\perp$.

The question is how to characterise the maximal isotropic subspaces of $\bf g$.

It is easy to see that the maximally isotropic subalgebras are precisely the Cartan subalgebras, but I am interested in subspaces which are not necessarily subalgebras.

The only examples I know are those for which $S = {\bf k}^\perp$ and ${\bf k} < {\bf g}$ a subalgebra, whence $${\bf g} = {\bf k} \oplus S$$ is a symmetric decomposition corresponding to the compact riemannian symmetric space $G/K$.

Question: Are there any other maximal isotropic subspaces?