Direct products and direct sums of Lie algebras

For a family $({\mathfrak{g}}_i{)}_{i\in I}$ of Lie algebras over $k$, the Cartesian product ${\prod }_i{\mathfrak{g}}_i$ has componentwise vector operations and bracket

$[(x_i),(y_i)]=([x_i,y_i]{)}_{i\in I}\mathrm{.}$

Bilinearity, alternation, and Jacobi hold componentwise, so this is the direct product Lie algebra. Its algebraic direct sum

${⨁}_{i\in I}{\mathfrak{g}}_i$

is the finite-support subspace of the product (The direct sum of an indexed family of modules) with the restricted bracket. It is closed because $\mathrm{supp}([(x_i),(y_i)])\subseteq \mathrm{supp}(x)\cap \mathrm{supp}(y)$, a finite set when both inputs have finite support.

For finite $I$, product and direct sum are the same Lie algebra. For $I=\mathrm{\varnothing }$, both are the zero Lie algebra. For two algebras the notation is $\mathfrak{g}\oplus \mathfrak{h}$ and the bracket is $[(x,u),(y,v)]=([x,y],[u,v])$.