Most of the answers can be found in Hochschild's book on the structure of Lie group.

a) Every complex semisimple group has a faithful rep (Thm 3.2 in Chap. XVII)

b) A connected Lie group with Levi decomposition $G=RS$ ($R$ the solvable radical, $S$ a semisimple Levi factor) is linear iff both $R$ and $S$ are linear (Thm 4.2 in Chap. XVIII)

c) A solvable Lie group $G$ is linear iff its commutator subgroup $G'$ is closed, and $G$ has no non-trivial compact subgroup (Thm 3.2 in Chap. XVIII)

Now, let $G$ be a semi-simple Lie group. Assume that $G$ is simply connected. Then $G$ admits a greatest linear quotient. Indeed, let $G_{\mathbb{C}}$ be the simply connected complex group corresponding to the complexified Lie algebra of $G$. Let $L$ be the kernel of the canonical homomorphism $G\rightarrow G_{\mathbb{C}}$; so $L$ is a finite index subgroup of the center of $G$. Then $G/L$ is the greatest linear quotient of $G$, in the sense that, if $H$ is locally isomorphic to $G$ and $p:G\rightarrow H$ is a universal covering, the group $H$ s linear iff $p$ factors through $G/L$.

Most of the answers can be found in Hochschild's book on the structure of Lie group.

a) Every complex semisimple group has a faithful rep (Thm 3.2 in Chap. XVII)

b) A connected Lie group with Levi decomposition $G=RS$ ($R$ the solvable radical, $S$ a semisimple Levi factor) is linear iff both $R$ and $S$ are linear (Thm 4.2 in Chap. XVIII)

c) A solvable Lie group $G$ is linear iff its commutator subgroup $G'$ is closed, and $G'$ has no non-trivial compact subgroup (Thm 3.2 in Chap. XVIII)

Now, let $G$ be a semi-simple Lie group. Assume that $G$ is simply connected. Then $G$ admits a greatest linear quotient. Indeed, let $G_{\mathbb{C}}$ be the simply connected complex group corresponding to the complexified Lie algebra of $G$. Let $L$ be the kernel of the canonical homomorphism $G\rightarrow G_{\mathbb{C}}$; so $L$ is a finite index subgroup of the center of $G$. Then $G/L$ is the greatest linear quotient of $G$, in the sense that, if $H$ is locally isomorphic to $G$ and $p:G\rightarrow H$ is a universal covering, the group $H$ s linear iff $p$ factors through $G/L$.

Most of the answers can be found in Hochschild's book on the structure of Lie group.

a) Every complex semisimple group has a faithful rep (Thm 3.2 in Chap. XVII)

b) A connected Lie group with Levi decomposition $G=RS$ ($R$ the solvable radical, $S$ a semisimple Levi factor) is linear iff both $R$ and $S$ are linear (Thm 4.2 in Chap. XVIII)

c) A solvable Lie group $G$ is linear iff its commutator subgroup $G'$ is closed, and $G$ has no non-trivial compact subgroup (Thm 3.2 in Chap. XVIII)

Now, let $G$ be a semi-simple Lie group. Assume that $G$ is simply connected. Then $G$ admits a greater linear quotient. Indeed, let $G_{\mathbb{C}}$ be the simply connected complex group corresponding to the complexified Lie algebra of $G$. Let $L$ be the kernel of the canonical homomorphism $G\rightarrow G_{\mathbb{C}}$; so $L$ is a finite index subgroup of the center of $G$. Then $G/L$ is the greatest linear quotient of $G$, in the sense that, if $H$ is locally isomorphic to $G$ and $p:G\rightarrow H$ is a universal covering, the group $H$ s linear iff $p$ factors through $G/L$.

Most of the answers can be found in Hochschild's book on the structure of Lie group.

a) Every complex semisimple group has a faithful rep (Thm 3.2 in Chap. XVII)

b) A connected Lie group with Levi decomposition $G=RS$ ($R$ the solvable radical, $S$ a semisimple Levi factor) is linear iff both $R$ and $S$ are linear (Thm 4.2 in Chap. XVIII)

c) A solvable Lie group $G$ is linear iff its commutator subgroup $G'$ is closed, and $G$ has no non-trivial compact subgroup (Thm 3.2 in Chap. XVIII)

Now, let $G$ be a semi-simple Lie group. Assume that $G$ is simply connected. Then $G$ admits a greatest linear quotient. Indeed, let $G_{\mathbb{C}}$ be the simply connected complex group corresponding to the complexified Lie algebra of $G$. Let $L$ be the kernel of the canonical homomorphism $G\rightarrow G_{\mathbb{C}}$; so $L$ is a finite index subgroup of the center of $G$. Then $G/L$ is the greatest linear quotient of $G$, in the sense that, if $H$ is locally isomorphic to $G$ and $p:G\rightarrow H$ is a universal covering, the group $H$ s linear iff $p$ factors through $G/L$.