TITLE: Minimal dimension of maximal abelian subalgebras
QUESTION [9 upvotes]: Let $\mathfrak{g}$ be a semi-simple finite dimensional  Lie algebra over $\mathbb C$.
Is it true that
$$
\dim \mathfrak h \ge \mathop{rank} \mathfrak g 
$$
for any maximal   abelian subalgebra $\mathfrak h \subset \mathfrak g$ ? Here by maximal I mean that $\mathfrak h$ is not properly contained in any other abelian subalgebra of $\mathfrak g$.

REPLY [3 votes]: Some of my previous comments were unfortunately too casual and unfocused, but I still suspect that the answer to the original question is yes.   At the same time, I can't document precisely enough what is known about the exceptional simple Lie algebras over $\mathbb{C}$.    I'm not aware of any unified Lie-theoretic arguments, in any case.
For the classical simple types (including the trace zero matrices?), the older papers are sometimes hard to read but have many concrete details.   The main tool used is linear algebra, inspired in part by a long tradition in the study of (associative) matrix algebras going back to Schur, Kravchuk, and Mal'cev.  In particular, there is a series of papers by Patera, Winternitz, and Zassenhaus on classical real or complex Lie algebras, some published in the journal Linear Algebra and its Applications which is in our university library but not available online to us.   One article I can access online is typical:  
MR713527 (84i:17006) 17B05 (22E10),
Patera, J. (3-MTRL-R); Winternitz, P. (3-MTRL-R); Zassenhaus, H. (1-OHS),
Maximal abelian subalgebras of real and complex symplectic Lie algebras.
J. Math. Phys. 24 (1983), no. 8, 1973–1985.
Their emphasis is on classifying the maximal ones up to conjugacy under the adjoint group, with emphasis on those like the Cartan subalgebras and the contrasting ones consisting of nilpotent matrices.   Dimension considerations seem secondary.    But as far as I can see, over $\mathbb{C}$ they always find that the minimum dimension of a maximal abelian subalgebra is equal to the rank, while the maximum dimension tends to be larger.   
Concerning jvp's own references to work of Gerstenhaber and others on the full matrix algebra, it has to be kept in mind that the simple linear Lie algebras in question aren't usually closed under ordinary matrix multiplication.   This makes comparisons of commutative subalgebras and abelian Lie algebras tricky, I think.   However, I haven't gone far enough into the literature to feel confident about exactly what is true.    It would help to have a modern survey of the entire problem area for both matrix algebras and semisimple Lie algebras.
ADDED: Though I haven't yet seen the Courter paper or his pre-computer proof method, the description here and jvp's reported computer verification suggests that for type A this Lie subalgebra of the special linear algebra consisting of nilpotent upper triangular matrices is maximal and of dimension less than the rank.   On the other hand, the published results for classical simple Lie algebras suggest that the rank is a lower bound in those types.   I'm uncertain how much is known about the exceptional types, so I guess I'd emphasize the last line of my earlier answer and the related question whether a Lie-theoretic understanding of the problem exists (beyond linear algebra).    But I'm happy to drop my original guess that the rank is always the minimum dimension if Courter's proof says otherwise.