A finite dimensional Lie algebra $\f$ is Frobenius if there is a linear Frobenius functional $F:\f \to \mathbb C$ such that the skew bilinear form $B_F$ defined by $B_F(x,y) = F([x,y])$ is non-degenerate. The principal element of $\f$ is then the unique element $\hat F$ such that $F(x) = F([\hat F, x])$; it depends on the choice of functional. However, if $\f$ is a subalgebra of a simple Lie algebra $\g$ and not an ideal of any larger subalgebra of $\g$ (in particular when $\f$ contains a Cartan subalgebra of $\g$) then the principal element is semisimple, and for $\g =\sl_n$ the eigenvalues of $\ad\hat F$ are shown to be integers which are independent of the choice of Frobenius functional. A basic open question is whether these eigenvalues characterize the algebra. The principal element of the first parabolic subalgebra of $\sl_n$ is shown to be the semisimple element of the principal three-dimensional subalgebra of $\sl_n$. Deformations of Frobenius Lie algebras remain Frobenius.