I ran into this question on math.stackexchange.com "How many 3 dimensional simple Lie algebras are there over the rationals?" The question has been sitting idle for a long time. I thought it was interesting and would like to know myself.

If working over $\mathbb{C}$, we know all 3-dimensional simple Lie algebras are isomorphic to $\mathfrak{sl}_2(\mathbb{C})$. When we move down to the reals, there are 2 non-isomorphic real forms.

The same argument that works over $\mathbb{C}$ works over $\bar{\mathbb{Q}}$ (the field of algebraic numbers). So over $\bar{\mathbb{Q}}$ the only 3-dimensional simple Lie algebra is $\mathfrak{sl}_2(\bar{\mathbb{Q}})$. Running through a similar argument as that which classifies 3-dim simples over $\mathbb{R}$, I get a whole mess of possibilities over $\mathbb{Q}$ (due to the lack of square roots) but have no idea which forms are non-isomorphic or even how to go about proving they are non-isomorphic.

Anybody know what the answer is? Have a good reference?

Thanks!

I ran into this question on math.stackexchange.com "How many 3 dimensional simple Lie algebras are there over the rationals?" The question has been sitting idle for a long time. I thought it was interesting and would like to know myself.

If working over $\mathbb{C}$, we know all 3-dimensional simple Lie algebras are isomorphic to $\mathfrak{sl}_2(\mathbb{C})$. When we move down to the reals, there are 2 non-isomorphic real forms.

The same argument that works over $\mathbb{C}$ works over $\bar{\mathbb{Q}}$ (the field of algebraic numbers). So over $\bar{\mathbb{Q}}$ the only 3-dimensional simple Lie algebra is $\mathfrak{sl}_2(\bar{\mathbb{Q}})$. Running through a similar argument as that which classifies 3-dim simples over $\mathbb{R}$, I get a whole mess of possibilities over $\mathbb{Q}$ (due to the lack of square roots) but have no idea which forms are non-isomorphic or even how to go about proving they are non-isomorphic.

Anybody know what the answer is? Have a good reference?

Thanks!