Show that $\mathfrak{b}_3 (\mathbb{C}) / \mathfrak{n}_3 (\mathbb{C}) \cong \mathfrak{t}_3 (\mathbb{C})$, in which $\mathfrak{b}_3 (\mathbb{C}),\mathfrak{n}_3 (\mathbb{C}),\mathfrak{t}_3 (\mathbb{C})$ are space all upper triangular matrices, space all strictly upper triangular matrices, diagonal matrices $3 \times 3$ respectively.
How to do this? I just "learn" Lie algebra for two days. Thank all!
Consider the map $\phi:\mathfrak{b}_3(\Bbb C)\to\mathfrak{t}_3(\Bbb C)$ which takes an upper triangular matrix $M$ to the diagonal matrix with the same diagonal entries. (i) Prove that $\phi$ is a homomorphism of associative algebras. (ii) Prove that $\phi$ is a homomorphism of Lie algebras. (iii) Prove its kernel is $\mathfrak{n}_3(\Bbb C)$. (iv) Apply the First Homomorphism Theorem for Lie algebras.
This works for other values of $3$ too.