# How many subalgebras are there in $sl_3$?

The Lie algebra $sl_3$ is 8 dimensional and $B=\{h_1, h_2, e_1, e_2, [e_1, e_2], f_1, f_2, [f_1, f_2]\}$ is a basis of $sl_3$. For every $x \in B$, $\text{Span}\{x\}$ is a one-dimensional subalgebra of $sl_3$. There are three 3-dimensional subalgebras of $sl_3$: $\text{Span}\{h_1, e_1, f_1\}$, $\text{Span}\{h_2, e_2, f_2\}$, $\text{Span}\{h_1+h_2, [e_1, e_2], [f_1, f_2]\}$. Fix $k_1, k_2 \in \mathbb{C}$, $\text{Span}\{k_1 h_1 + k_2 h_2\}$ is also a 1-dimensional subalgebra of $sl_3$. The algebras $\text{Span}\{h_1, e_1\}$, $\text{Span}\{h_1, e_2\}$ are 2-dimensional subalgebras of $sl_3$. My question is: are $\text{Span}\{h_1, e_1, f_1\}$, $\text{Span}\{h_2, e_2, f_2\}$, $\text{Span}\{h_1+h_2, [e_1, e_2], [f_1, f_2]\}$ maximal subalgebras of $sl_3$? How to write down all maximal subalgebras of $sl_3$? Thank you very much.

• Dynkin classified all maximal subalgebras of simple Lie algebras in his $1952$ paper. You can use sage to list them. – Dietrich Burde Mar 13 '16 at 9:52
• @Dietrich Burde, thank you very much. I searched on google but didn't find the commands of sage which will return the maximal Lie subalgebras of simple Lie algebras. Do you know the commands of sage which will list all maximal subalgebras or some references about this? – LJR Mar 13 '16 at 10:18
• A references is this paper; GAP4 is also part of sage. – Dietrich Burde Mar 13 '16 at 14:41
• Maximal subalgebra are either irreducible, or subspace stabilizers. In particular, your $\mathrm{Span}\{h_i,e_i,f_i\}$ are not irreducible and hence not maximal (and with little effort you can explicitly write down bases for larger subalgebras). – YCor Mar 13 '16 at 16:30
• No, it's not irreducible (by irreducible, I mean it acts irreducibly on the 3-dimensional space). – YCor Mar 14 '16 at 9:55