Rather than an answer, this is more of an anti-answer: I'll try to persuade you that you probably don't want to know the answer to your question.

Instead of some exotic nonalgebraic Lie algebra, let's start with the one-dimensional Lie algebra $g$ over a field k. A representation of $g$ is just a finite-dimensional vector space + an endomorphism. I don't know what the affine group scheme attached to this Tannakian category is, but thanks to a 1954 paper of Iwahori, I can tell you that its Lie algebra can be identified with the set of pairs $(g,c)$ where $g$ is a homomorphism of abelian groups $k\to k$ and $c$ is an element of $k$. So if $k$ is big, this is huge; in particular, you don't get an algebraic group. (Added: $k$ is algebraically closed.)

By contrast, the affine group scheme attached to the category of representations of a semisimple Lie algebra g in characteristic zero is the simply connected algebraic group with Lie algebra g.

In summary: this game works beautifully for semisimple Lie algebras (in characteristic zero), but otherwise appears to be a big mess. See arXiv:0705.1348 for a few more details.

Rather than an answer, this is more of an anti-answer: I'll try to persuade you that you probably don't want to know the answer to your question.

Instead of some exotic nonalgebraic Lie algebra, let's start with the one-dimensional Lie algebra $\mathfrak{g}$ over a field $k$. A representation of $\mathfrak{g}$ is just a finite-dimensional vector space + an endomorphism. I don't know what the affine group scheme attached to this Tannakian category is, but thanks to a 1954 paper of Iwahori, I can tell you that its Lie algebra can be identified with the set of pairs $(\mathfrak{g},c)$ where $\mathfrak{g}$ is a homomorphism of abelian groups $k\to k$ and $c$ is an element of $k$. So if $k$ is big, this is huge; in particular, you don't get an algebraic group. (Added: $k$ is algebraically closed.)

By contrast, the affine group scheme attached to the category of representations of a semisimple Lie algebra $\mathfrak{g}$ in characteristic zero is the simply connected algebraic group with Lie algebra $\mathfrak{g}$.

In summary: this game works beautifully for semisimple Lie algebras (in characteristic zero), but otherwise appears to be a big mess. See arXiv:0705.1348 for a few more details.

Rather than an answer, this is more of an anti-answer: I'll try to persuade you that you probably don't want to know the answer to your question.

Instead of some exotic nonalgebraic Lie algebra, let's start with the one-dimensional Lie algebra $g$ over a field k. A representation of $g$ is just a finite-dimensional vector space + an endomorphism. I don't know what the affine group scheme attached to this Tannakian category is, but thanks to a 1954 paper of Iwahori, I can tell you that its Lie algebra can be identified with the set of pairs $(g,c)$ where $g$ is a homomorphism of abelian groups $k\to k$ and $c$ is an element of $k$. So if $k$ is big, this is huge; in particular, you don't get an algebraic group.

By contrast, the affine group scheme attached to the category of representations of a semisimple Lie algebra g in characteristic zero is the simply connected algebraic group with Lie algebra g.

In summary: this game works beautifully for semisimple Lie algebras, but otherwise appears to be a big mess. See arXiv:0705.1348 for a few more details.

Rather than an answer, this is more of an anti-answer: I'll try to persuade you that you probably don't want to know the answer to your question.

Instead of some exotic nonalgebraic Lie algebra, let's start with the one-dimensional Lie algebra $g$ over a field k. A representation of $g$ is just a finite-dimensional vector space + an endomorphism. I don't know what the affine group scheme attached to this Tannakian category is, but thanks to a 1954 paper of Iwahori, I can tell you that its Lie algebra can be identified with the set of pairs $(g,c)$ where $g$ is a homomorphism of abelian groups $k\to k$ and $c$ is an element of $k$. So if $k$ is big, this is huge; in particular, you don't get an algebraic group. (Added: $k$ is algebraically closed.)

By contrast, the affine group scheme attached to the category of representations of a semisimple Lie algebra g in characteristic zero is the simply connected algebraic group with Lie algebra g.

In summary: this game works beautifully for semisimple Lie algebras (in characteristic zero), but otherwise appears to be a big mess. See arXiv:0705.1348 for a few more details.