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There are exactly four non-solvable complex Lie algebras of dimension $6$ with nontrivial Levi decomposition, namely \begin{align*} \mathfrak{sl}_2(\Bbb C) & \ltimes_{\phi} \Bbb C^3\\ \mathfrak{sl}_2(\Bbb C) & \ltimes_{\phi}\mathfrak{r}_{3,1}(\Bbb C) \cong \mathfrak{aff}(\Bbb C^2)\cong \mathfrak{gl}_2(\Bbb C)\ltimes \Bbb C^2\\ \mathfrak{sl}_2(\Bbb C) & \ltimes_{\phi} \mathfrak{n}_3(\Bbb C)\\ (\mathfrak{sl}_2(\Bbb C) &\ltimes_{\phi} \Bbb C^2))\oplus \Bbb C \end{align*} The proof is straightforward, by computing all derivation algebras of $3$-dimensional Lie algebras $\mathfrak{r}$, and then considering Lie algebra homomorphisms $$ \phi\colon \mathfrak{sl}_2(\Bbb C)\rightarrow {\rm Der}(\mathfrak{r}). $$ The only derivation Lie algebras with are not solvable are the ones of $\Bbb C^3$, of the Heisenberg Lie algebra $\mathfrak{n}_3(\Bbb C)$, and of the solvable non-nilpotent Lie algebra $\mathfrak{r}_{3,1}(\Bbb C)$. Their derivation algebras are isomorphic to $\mathfrak{sl}_3(\Bbb C)\oplus \Bbb C$, $\mathfrak{aff}(\Bbb C^2)$ and $\mathfrak{aff}(\Bbb C^2)$.

I should add one reference, for the case of real numbers, namely the classification of all solvable real six-dimensional Lie algebras by Turkowsky:

P. Turkowski. Solvable Lie algebras of dimension six. J. Math. Phys. 31 (1990), 1344-1350.

As mentioned before, the classification of nilpotent Lie algebras of dimension $6$ has been done earlier by Morozow in $1958$, valid for all fields of characteristic zero. Recently this classification has been extended to arbitrary fields, by Serena Cicalo, Willem A de Graaf and Csaba Schneider here:http://arxiv.org/abs/1011.0361.

There are exactly four non-solvable complex Lie algebras of dimension $6$ with nontrivial Levi decomposition, namely \begin{align*} \mathfrak{sl}_2(\Bbb C) & \ltimes_{\phi} \Bbb C^3\\ \mathfrak{sl}_2(\Bbb C) & \ltimes_{\phi}\mathfrak{r}_{3,1}(\Bbb C) \cong \mathfrak{aff}(\Bbb C^2)\cong \mathfrak{gl}_2(\Bbb C)\ltimes \Bbb C^2\\ \mathfrak{sl}_2(\Bbb C) & \ltimes_{\phi} \mathfrak{n}_3(\Bbb C)\\ (\mathfrak{sl}_2(\Bbb C) &\ltimes_{\phi} \Bbb C^2))\oplus \Bbb C \cong \mathfrak{sl}_2(\Bbb C)\ltimes_{\psi}\Bbb C^3 \end{align*} The proof is straightforward, by computing all derivation algebras of $3$-dimensional Lie algebras $\mathfrak{r}$, and then considering Lie algebra homomorphisms $$ \phi\colon \mathfrak{sl}_2(\Bbb C)\rightarrow {\rm Der}(\mathfrak{r}). $$ The only derivation Lie algebras with are not solvable are the ones of $\Bbb C^3$, of the Heisenberg Lie algebra $\mathfrak{n}_3(\Bbb C)$, and of the solvable non-nilpotent Lie algebra $\mathfrak{r}_{3,1}(\Bbb C)$. Their derivation algebras are isomorphic to $\mathfrak{sl}_3(\Bbb C)\oplus \Bbb C$, $\mathfrak{aff}(\Bbb C^2)$ and $\mathfrak{aff}(\Bbb C^2)$.

I should add one reference, for the case of real numbers, namely the classification of all solvable real six-dimensional Lie algebras by Turkowsky:

P. Turkowski. Solvable Lie algebras of dimension six. J. Math. Phys. 31 (1990), 1344-1350.

As mentioned before, the classification of nilpotent Lie algebras of dimension $6$ has been done earlier by Morozow in $1958$, valid for all fields of characteristic zero. Recently this classification has been extended to arbitrary fields, by Serena Cicalo, Willem A de Graaf and Csaba Schneider here:http://arxiv.org/abs/1011.0361.

There are exactly three non-solvable complex Lie algebras of dimension $6$ with nontrivial Levi decomposition, namely \begin{align*} \mathfrak{sl}_2(\Bbb C) & \ltimes_{\phi} \Bbb C^3\\ \mathfrak{sl}_2(\Bbb C) & \ltimes_{\phi}\mathfrak{r}_{3,1}(\Bbb C) \cong \mathfrak{aff}(\Bbb C^2)\cong \mathfrak{gl}_2(\Bbb C)\ltimes \Bbb C^2\\ \mathfrak{sl}_2(\Bbb C) & \ltimes_{\phi} \mathfrak{n}_3(\Bbb C).\\ \end{align*} The proof is straightforward, by computing all derivation algebras of $3$-dimensional Lie algebras $\mathfrak{r}$, and then considering Lie algebra homomorphisms $$ \phi\colon \mathfrak{sl}_2(\Bbb C)\rightarrow {\rm Der}(\mathfrak{r}). $$ The only derivation Lie algebras with are not solvable are the ones of $\Bbb C^3$, of the Heisenberg Lie algebra $\mathfrak{n}_3(\Bbb C)$, and of the solvable non-nilpotent Lie algebra $\mathfrak{r}_{3,1}(\Bbb C)$. Their derivation algebras are isomorphic to $\mathfrak{sl}_3(\Bbb C)\oplus \Bbb C$, $\mathfrak{aff}(\Bbb C^2)$ and $\mathfrak{aff}(\Bbb C^2)$.

I should add one reference, for the case of real numbers, namely the classification of all solvable real six-dimensional Lie algebras by Turkowsky:

P. Turkowski. Solvable Lie algebras of dimension six. J. Math. Phys. 31 (1990), 1344-1350.

As mentioned before, the classification of nilpotent Lie algebras of dimension $6$ has been done earlier by Morozow in $1958$, valid for all fields of characteristic zero. Recently this classification has been extended to arbitrary fields, by Serena Cicalo, Willem A de Graaf and Csaba Schneider here:http://arxiv.org/abs/1011.0361.

There are exactly four non-solvable complex Lie algebras of dimension $6$ with nontrivial Levi decomposition, namely \begin{align*} \mathfrak{sl}_2(\Bbb C) & \ltimes_{\phi} \Bbb C^3\\ \mathfrak{sl}_2(\Bbb C) & \ltimes_{\phi}\mathfrak{r}_{3,1}(\Bbb C) \cong \mathfrak{aff}(\Bbb C^2)\cong \mathfrak{gl}_2(\Bbb C)\ltimes \Bbb C^2\\ \mathfrak{sl}_2(\Bbb C) & \ltimes_{\phi} \mathfrak{n}_3(\Bbb C)\\ (\mathfrak{sl}_2(\Bbb C) &\ltimes_{\phi} \Bbb C^2))\oplus \Bbb C \end{align*} The proof is straightforward, by computing all derivation algebras of $3$-dimensional Lie algebras $\mathfrak{r}$, and then considering Lie algebra homomorphisms $$ \phi\colon \mathfrak{sl}_2(\Bbb C)\rightarrow {\rm Der}(\mathfrak{r}). $$ The only derivation Lie algebras with are not solvable are the ones of $\Bbb C^3$, of the Heisenberg Lie algebra $\mathfrak{n}_3(\Bbb C)$, and of the solvable non-nilpotent Lie algebra $\mathfrak{r}_{3,1}(\Bbb C)$. Their derivation algebras are isomorphic to $\mathfrak{sl}_3(\Bbb C)\oplus \Bbb C$, $\mathfrak{aff}(\Bbb C^2)$ and $\mathfrak{aff}(\Bbb C^2)$.

I should add one reference, for the case of real numbers, namely the classification of all solvable real six-dimensional Lie algebras by Turkowsky:

P. Turkowski. Solvable Lie algebras of dimension six. J. Math. Phys. 31 (1990), 1344-1350.

As mentioned before, the classification of nilpotent Lie algebras of dimension $6$ has been done earlier by Morozow in $1958$, valid for all fields of characteristic zero. Recently this classification has been extended to arbitrary fields, by Serena Cicalo, Willem A de Graaf and Csaba Schneider here:http://arxiv.org/abs/1011.0361.

I should add one reference, for the case of real numbers, namely the classification of all solvable real six-dimensional Lie algebras by Turkowsky:

P. Turkowski. Solvable Lie algebras of dimension six. J. Math. Phys. 31 (1990), 1344-1350.

As mentioned before, the classification of nilpotent Lie algebras of dimension $6$ has been done earlier by Morozow in $1958$, valid for all fields of characteristic zero. Recently this classification has been extended to arbitrary fields, by Serena Cicalo, Willem A de Graaf and Csaba Schneider here:http://arxiv.org/abs/1011.0361.

There are exactly three non-solvable complex Lie algebras of dimension $6$ with nontrivial Levi decomposition, namely \begin{align*} \mathfrak{sl}_2(\Bbb C) & \ltimes_{\phi} \Bbb C^3\\ \mathfrak{sl}_2(\Bbb C) & \ltimes_{\phi}\mathfrak{r}_{3,1}(\Bbb C) \cong \mathfrak{aff}(\Bbb C^2)\cong \mathfrak{gl}_2(\Bbb C)\ltimes \Bbb C^2\\ \mathfrak{sl}_2(\Bbb C) & \ltimes_{\phi} \mathfrak{n}_3(\Bbb C).\\ \end{align*} The proof is straightforward, by computing all derivation algebras of $3$-dimensional Lie algebras $\mathfrak{r}$, and then considering Lie algebra homomorphisms $$ \phi\colon \mathfrak{sl}_2(\Bbb C)\rightarrow {\rm Der}(\mathfrak{r}). $$ The only derivation Lie algebras with are not solvable are the ones of $\Bbb C^3$, of the Heisenberg Lie algebra $\mathfrak{n}_3(\Bbb C)$, and of the solvable non-nilpotent Lie algebra $\mathfrak{r}_{3,1}(\Bbb C)$. Their derivation algebras are isomorphic to $\mathfrak{sl}_3(\Bbb C)\oplus \Bbb C$, $\mathfrak{aff}(\Bbb C^2)$ and $\mathfrak{aff}(\Bbb C^2)$.

I should add one reference, for the case of real numbers, namely the classification of all solvable real six-dimensional Lie algebras by Turkowsky:

P. Turkowski. Solvable Lie algebras of dimension six. J. Math. Phys. 31 (1990), 1344-1350.

As mentioned before, the classification of nilpotent Lie algebras of dimension $6$ has been done earlier by Morozow in $1958$, valid for all fields of characteristic zero. Recently this classification has been extended to arbitrary fields, by Serena Cicalo, Willem A de Graaf and Csaba Schneider here:http://arxiv.org/abs/1011.0361.