sl(2,R) direct sum with itself

The direct-sum real Lie algebra $\mathfrak{sl}(2,\mathbb R)\oplus \mathfrak{sl}(2,\mathbb R)$ has ordered pairs of elements of $\mathfrak{sl}(2,\mathbb R)$ as its vectors, with componentwise scalar operations and bracket

[(x_1,x_2),(y_1,y_2)]=([x_1,y_1],[x_2,y_2]).

Two commuting ideals

The two coordinate summands are commuting ideals of dimension three. The direct sum has dimension six over its stated scalar field and is semisimple but not simple. This product decomposition is part of the presented object, while its isomorphic orthogonal presentation uses a different matrix carrier.