Simple Lie algebra representations and tensor powers of fundamental representations

Question

Let $\frak{g}$ be a simple Lie algebra over $\mathbb{C}$. We will call a representation of $\frak{g}$ tautological if it is a fundamental representation of smallest dimension.

For $V$ a tautological representation, can all representations of $\frak{g}$ be realised as a sub-representation of some tensor power of $V$?

My intuiton here comes from the A-series . . .