Above, "$\binom{n}{k}$ for $n \geq 2$" should be "$\binom{n}{k}$ for $2 \leq k \leq n-2$."

Among the exceptional Lie algebras, the only one with dimension being a binomial coefficient $\binom {n}{k}$ for $n \geq 2$ is $E_6$ with $78 = \binom{13}{2}$. However, the smallest non-trivial rep. of $E_6$ has dimension 27, which rules out 13. Among the classical Lie algebras, as $sl(n)$-modules, $sl(n)$ is essentially iso. to $I^{\otimes 2}$ (with the trivial rep. deleted), and as $sp(n)$-modules, $sp(n)$ is iso. to the second symmetric power of $I$. For $k \geq 2$ and $g$ complex, simple, the only solutions to the question are from the $so(n)$ family.

Above, $I^2$ denotes $I^{\wedge 2}$.

As an example of this situation, every element of the orthogonal Lie algebra $so(n)$ can be written as an $n \times n$ matrix, which then acts on ${\bf R}^n$ via matrix multiplication on vectors. Here $I = {\bf R}^n$ is "the standard representation." It can be easily checked that as $so(n)$-modules, $so(n) \simeq I^2$. Additionally, $I$ can be viewed as an Abelian Lie algebra (all brackets zero), and $h$ contains all linear combinations of elements in $so(n)$ with elements in $I$.