Rectangular representations and $\lambda$-independence of algebraic monodromy groups

Let $\mathfrak{g}$ be a complex semisimple Lie algebra. We define what it means for a finite dimensional representation of $\mathfrak{g}$ to be rectangular and completely classify faithful rectangular representations. As an application, we obtain new $\lambda$-independence results on the algebraic monodromy groups of compatible systems of $\lambda$-adic Galois representations of number fields.