Are there proper Ad invariant sets on simple lie algebras?

Question

Let $\mathcal C\subset \mathfrak g$ be a subset in a Lie algebra $\mathfrak g$ satisfying the following two conditions:

1. $\mathrm{Ad}(G)\mathcal C=\mathcal C$
2. If $X,Y\in \mathcal C$, then $X+Y\in\mathcal C$ or $X-Y\in\mathcal C$.

(one can also assume 3. There is a proper subspace $\mathcal H\subset \mathfrak g$ such that $\mathcal H\subset \mathcal C$)

Is it true that $\mathcal C=\mathfrak g$?

By $\mathrm{Ad}$ I mean the map induced by the derivative of the conjugation at the identity on a Lie group.