A converse of Whitehead's first and second lemma has been recently studied by P. Zusmanovich, e.g., see here. One of the results is as follows:

Theorem 0.2 (A converse to the Second Whitehead Lemma). A finite-dimensional Lie algebra over a field of characteristic zero such that its second cohomology with coefficients in any finite-dimensional module vanishes, is one of the following:

(i) an one-dimensional algebra;
(ii) a semisimple algebra;
(iii) the direct sum of a semisimple algebra and an one-dimensional algebra.

If you only require that $H^2$ vanishes for the trivial representation, we have much less, of course. By Dixmier's result, such a Lie algebra cannot be nilpotent. More details are in Zusmanovich's paper.

A converse of Whitehead's first and second lemma has been recently studied by P. Zusmanovich, e.g., see here. One of the results is as follows:

Theorem 0.2 (A converse to the Second Whitehead Lemma). A finite-dimensional Lie algebra over a field of characteristic zero such that its second cohomology with coefficients in any finite-dimensional module vanishes, is one of the following:

(i) an one-dimensional algebra;
(ii) a semisimple algebra;
(iii) the direct sum of a semisimple algebra and an one-dimensional algebra.

If you only require that $H^2$ vanishes for the trivial representation, we have much less, of course. By Dixmier's result, such a Lie algebra cannot be nilpotent. It might be solvable, however, e.g., a solvable, non-nilpotentent strongly rigid Lie algebra. More details are in Zusmanovich's paper.

The converse of Whitehead's first and second lemma has been recently studied by P. Zusmanovich, e.g., see here. One of the results is as follows:

Theorem 0.2 (A converse to the Second Whitehead Lemma). A finite-dimensional Lie algebra over a field of characteristic zero such that its second cohomology with coefficients in any finite-dimensional module vanishes, is one of the following:

(i) an one-dimensional algebra;
(ii) a semisimple algebra;
(iii) the direct sum of a semisimple algebra and an one-dimensional algebra.

If you only require that $H^2$ vanishes for the trivial representation, we have much less, of course.

A converse of Whitehead's first and second lemma has been recently studied by P. Zusmanovich, e.g., see here. One of the results is as follows:

Theorem 0.2 (A converse to the Second Whitehead Lemma). A finite-dimensional Lie algebra over a field of characteristic zero such that its second cohomology with coefficients in any finite-dimensional module vanishes, is one of the following:

(i) an one-dimensional algebra;
(ii) a semisimple algebra;
(iii) the direct sum of a semisimple algebra and an one-dimensional algebra.

If you only require that $H^2$ vanishes for the trivial representation, we have much less, of course. By Dixmier's result, such a Lie algebra cannot be nilpotent. More details are in Zusmanovich's paper.