A Lie subalgebra of gl(n,k) which is the Lie algebra of an algebraic subgroup of GL(n,k) is called an algebraic subalgebra. Apparently there are Lie subalgebras which are not algebraic, even in characteristic zero. If g is the Lie algebra of an affine algebraic group then it must be ad-algebraic, ie. its image in End(g) under the adjoint representation must be an algebraic subalgebra. An example of a non-ad-algebraic Lie algebra is given on pg. 385 of Lie Algebras and Algebraic Groups, by Tauvel and Yu.

A Lie subalgebra of $\mathfrak{gl}(n,k)$ which is the Lie algebra of an algebraic subgroup of $GL(n,k)$ is called an algebraic subalgebra. Apparently there are Lie subalgebras which are not algebraic, even in characteristic zero. If $\mathfrak{g}$ is the Lie algebra of an affine algebraic group then it must be ad-algebraic, ie. its image in $\operatorname{End}(\mathfrak{g})$ under the adjoint representation must be an algebraic subalgebra. An example of a non-ad-algebraic Lie algebra is given on pg. 385 of Lie Algebras and Algebraic Groups, by Tauvel and Yu.

A Lie subalgebra of gl(n,k) which is the Lie algebra of an algebraic subgroup of GL(n,k) is called an algebraic subalgebra. Apparently there are Lie subalgebras which are not algebraic, even in characteristic zero. If g is the Lie algebra of an affine algebraic group then it must be ad-algebraic, ie. its image in End(g) under the adjoint representation must be an algebraic subalgebra. An example of a non-ad-algebraic Lie algebra is given on pg. 385 of Lie Algebras and Algebraic Groups, by Tauvel and Yu.

A Lie subalgebra of gl(n,k) which is the Lie algebra of an algebraic subgroup of GL(n,k) is called an algebraic subalgebra. Apparently there are Lie subalgebras which are not algebraic, even in characteristic zero. If g is the Lie algebra of an affine algebraic group then it must be ad-algebraic, ie. its image in End(g) under the adjoint representation must be an algebraic subalgebra. An example of a non-ad-algebraic Lie algebra is given on pg. 385 of Lie Algebras and Algebraic Groups, by Tauvel and Yu.