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The general linear group $GL(n,\mathbb{R})$ is the set of $(n\times n)$ invertible matrices.

How can its Lie algebra $\mathfrak{gl}(n,\mathbb{R})$ be written in terms of simple Lie algebras (ABCDE), as classified by Cartan and possibly other factors?

For example, a subgroup of the general linear group is the Lorentz group and the corresponding Lie algebra can be written as $\mathfrak{sl}(2,\mathbb{C}) \simeq A_1 \times A_1 $ .

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We have $\mathfrak{gl}_n(\mathbb{R})\cong \mathfrak{sl}_n(\mathbb{R})\oplus \mathbb{R}$, which is the direct sum of an algebra of type $A_{n-1}$ and a $1$-dimensional abelian Lie algebra (which is not simple by convention).

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