A reference is "Lectures on Real Semisimple Lie Algebras and Their Representations" by A. L. Onishchik. A first result here is: any irreducible real representation $\rho\colon \mathbb{g}\rightarrow \mathbb{gl}(V)$ of a real Lie algebra $\mathbb{g}$ satisfies precisely one of the following two conditions.
$(1)$ $\rho^{\mathbb{C}}$ is an irreducible complex representation.
$(2)$ $\rho=\rho'_{\mathbb{R}}$, where $\rho'$ is an irreducible complex representation admitting no invariant real structure.
Conversely, any real representation satisfying $(1)$ or $(2)$ is irreducible.

For more results, concerning highest weigts etc. see Onishchik's book, section $8$, about "Real representations of real semisimple Lie algebras", including the classification of them.

A reference for this is also "Lectures on Real Semisimple Lie Algebras and Their Representations" by A. L. Onishchik. A first result here is: any irreducible real representation $\rho\colon \mathbb{g}\rightarrow \mathbb{gl}(V)$ of a real Lie algebra $\mathbb{g}$ satisfies precisely one of the following two conditions.
$(1)$ $\rho^{\mathbb{C}}$ is an irreducible complex representation.
$(2)$ $\rho=\rho'_{\mathbb{R}}$, where $\rho'$ is an irreducible complex representation admitting no invariant real structure.
Conversely, any real representation satisfying $(1)$ or $(2)$ is irreducible.

For the results concerning highest weigts etc. see Onishchik's book, section $8$, about "Real representations of real semisimple Lie algebras", including the classification of them.

A reference is "Lectures on Real Semisimple Lie Algebras and Their Representations" by A. L. Onishchik. A first result here is: any irreducible real representation $\rho\colon \mathbb{g}\rightarrow \mathbb{gl}(V)$ of a real Lie algebra $\mathbb{g}$ satisfies precisely one of the following two conditions.
$(1)$ $\rho^{\mathbb{C}}$ is an irreducible complex representation.
$(2)$ $\rho=\rho'_{\mathbb{R}}$, where $\rho'$ is an irreducible complex representation admitting no invariant real structure.
Conversely, any real representation satisfying $(1)$ or $(2)$ is irreducible.

For more results, concerning highest weigts etc. see Onishchik's book.

A reference is "Lectures on Real Semisimple Lie Algebras and Their Representations" by A. L. Onishchik. A first result here is: any irreducible real representation $\rho\colon \mathbb{g}\rightarrow \mathbb{gl}(V)$ of a real Lie algebra $\mathbb{g}$ satisfies precisely one of the following two conditions.
$(1)$ $\rho^{\mathbb{C}}$ is an irreducible complex representation.
$(2)$ $\rho=\rho'_{\mathbb{R}}$, where $\rho'$ is an irreducible complex representation admitting no invariant real structure.
Conversely, any real representation satisfying $(1)$ or $(2)$ is irreducible.

For more results, concerning highest weigts etc. see Onishchik's book, section $8$, about "Real representations of real semisimple Lie algebras", including the classification of them.