Exercise 9, § 7

Tag 014L, permanent. Cite it as [Bourbaki, Tag 014L].

Lie Groups and Lie Algebras, VIII, 241

Let $\mathfrak{a}_1$ and $\mathfrak{a}_2$ be Lie algebras. Show that there exists a unique homomorphism

f: R(\mathfrak{a}_1)\otimes_{\mathbf{Z}}R(\mathfrak{a}_2)\rightarrow R(\mathfrak{a}_1\times \mathfrak{a}_2)

such that $f([E_1]\otimes [E_2]) = [E_1\otimes E_2]$ if $E_i$ is an $\mathfrak{a}_i$ -module $(i= 1,2)$ . Show that $f$ is injective, and that it is bijective if $\mathfrak{a}_1$ and $\mathfrak{a}_2$ are splittable semi-simple.

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