  We give a characterization of symplectic quadratic Lie algebras that their
Lie algebra of inner derivations has an invertible derivation. A family of
symplectic quadratic Lie algebras is introduced to illustrate this situation.
Finally, we calculate explicitly the Betti numbers of a family of solvable Lie
algebras in two ways: using the cohomology of quadratic Lie algebras and
applying a Pouseele's result on extensions of the one-dimensional Lie algebra
by Heisenberg Lie algebras
