  In this paper, we present a modern version of Gordan's algorithm on binary
forms. Symbolic method is reinterpreted in terms of
$\mathsf{SL}_2(\mathbb{C})$--equivariant homomorphisms defined upon Cayley
operator and polarization operator. A graphical approach is thus developed to
obtain Gordan's ideal, a central key to get covariant bases of binary forms. To
illustrate the power of this method, we obtain for the first time a minimal
covariant basis for $\Sn{6}\oplus\Sn{4}$ and $\Sn{6}\oplus\Sn{4}\oplus \Sn{2}$.

Representation Theory;Commutative Algebra