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Let $L$ be a simple Lie algebra over ${\rm GF}(2)$. If $α$ is an automorphism of $L$ then for any element of $L$ we must have $α[a,b]=[α(a),α(b)]$. Now I want to have a clear understanding of image of $α$. Since $L$ is a simple Lie algebra, how can I write the image of $α$? I think it is not true to write the image of $α$ as a linear combination of basis. In fact, my confusion backs to the notion of structure constant of a Lie algebra. I know that we must consider Lie bracket and structure constants, but I need examples.

Ivo Terek
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user146566
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    An automorphism is also a bijective linear map $\phi\colon L\rightarrow L$ of the underlying vector space. So the image is $L$ as a vector space, since $\phi$ is surjective. – Dietrich Burde Jun 19 '14 at 14:46
  • Even if you are only interested in enomorphism: IIRC the kernel a Lie algebra homomorphism is an ideal. If $L$ is simple, then... – Jyrki Lahtonen Jun 19 '14 at 15:33

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