Invariant Adjoint Tensors of the Classical Groups

For $\mathfrak{g}$ a simple Lie algebra and $G$ its adjoint group, the Chevalley map and work of Coxeter gives a concrete description of the algebra of $G$-invariant polynomials on $\mathfrak{g}$ in terms of traces over various representations. Here we provide an extension of this description to $G$-invariant tensors on $\mathfrak{g}$, although restricted to only providing generators and only for the classical Lie algebras.