# Automorphisms of direct sum of Lie algebras

Let $\mathfrak{g}$ be a (finite-dimensional, linear) Lie algebra. Is it always true that $$\mathrm{Aut}(\mathfrak{g\oplus g})\supset\mathrm{Aut}\mathfrak g\times \mathrm{GL_2}\mathbb R$$

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What do you mean by $\operatorname{Aut}(\mathfrak g)\otimes \operatorname{Aut}(\mathfrak h)$? You cannot compute tensor products of what are generally non-abelian groups. –  Mariano Suárez-Alvarez Apr 18 at 4:25
And what do you mean by $\mathfrak{g} \otimes \mathfrak{h}$? AFAIK there is no notion of tensor product for Lie algebras. –  Qiaochu Yuan Apr 18 at 4:26
@MarianoSuárez-Alvarez I see. My bad. I found this though. –  Earthling Apr 18 at 4:33
@user1205935, that has no relevance here, as you'll immediately see if you try to apply to the context of the question you originally wrote. You need a pair of compatible actions and so on: your question still does not make sense, I am afraid. (You should also fix the title, btw) –  Mariano Suárez-Alvarez Apr 18 at 4:42
Back to the blackboard, then. Thank you for your comments. –  Earthling Apr 18 at 4:46
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