Let $L$ be a finite dimensional semisimple complex Lie algebra.
Let $M$ be a subalgebra with the property that all elements of $M$ are semisimple, and is maximal w.r.t. this property.
Then $M$ is abelian and its centralizer is itself. So it is maximal abelian subalgebra.
Q.1 Is it always true that a maximal abelian subalgebra should contain only semisimple elements?
Q.2 Is it true that two maximal abelian subalgebras can have different dimensions? (I want to know answer to Q. 2 in which one may make cases - $L$ is emisimple or not semisimple. )