Semisimple algebras and disjoint unions of diagrams

The simple-ideal decomposition is unique up to order. Indeed, given decompositions $\mathfrak{g}={⨁}_{i=1}^m{\mathfrak{g}}_i={⨁}_{j=1}^n{\mathfrak{h}}_j$, [L3] writes each ideal ${\mathfrak{g}}_i$ as a sum of a subfamily of the ${\mathfrak{h}}_j$. Simplicity and nonzeroness force that subfamily to consist of exactly one factor, so ${\mathfrak{g}}_i={\mathfrak{h}}_j$ for a unique $j$. Distinct $i$ give distinct $j$, and every ${\mathfrak{h}}_j$ occurs because the ${\mathfrak{g}}_i$ span $\mathfrak{g}$. Thus $m=n$ and the two families agree after a permutation.

Conversely, a finite multiset of connected finite-type diagrams determines a semisimple algebra up to isomorphism: take the direct sum of the simple Lie algebras attached to the diagrams by [L2]; any two semisimple algebras with the same multiset of simple-ideal diagrams are isomorphic factor by factor by [L2].

By [L1] write $\mathfrak{g}={\mathfrak{g}}_1\oplus \cdots \oplus {\mathfrak{g}}_m$ as a direct sum of simple ideals. Each ${\mathfrak{g}}_j$ is a finite-dimensional complex simple Lie algebra, so by [L2] it has a connected finite-type Dynkin diagram, well defined up to isomorphism, and step 1.1 makes the resulting multiset depend only on the isomorphism class of $\mathfrak{g}$.

Steps 1.2 and 2.1 give inverse assignments between isomorphism classes of finite-dimensional complex semisimple Lie algebras and finite multisets of connected finite-type Dynkin diagrams, which is the asserted classification with multiplicity.