Consider two (possibly infinite-dimensional) representations $\rho$, $\pi$ of a semisimple Lie algebra $\mathfrak{g}$, with $\rho$ irreducible and $\pi$ indecomposable but not irreducible (i.e., not simple). Are there any circumstances under which the product representation $\rho\otimes\pi$ can be semisimple (isomorphic to a direct sum of irreducible representations)?

In other words: is the product of a simple and a non-semisimple representation always non-semisimple?

Motivation

I'm working in particular with $\mathfrak{spin}(3,1)$. The situation is the following: I have a class of irreducible finite-dimensional representations (the $\rho$ above) that I know how to couple with generic irreducible representations $\sigma$ (that is, I know under which conditions the resulting representation is semisimple or not).
I need to find out if the product $\rho_1\otimes\rho_2\otimes\sigma$ is semisimple: if $\rho_2\otimes\sigma$ is, then I can consider the product with $\rho_1$ for each one of the irreducibles in the decomposition to obtain the answer; if, however, $\rho_2\otimes\sigma$ is indecomposable but not irreducible, can I be sure that $\rho_1\otimes\rho_2\otimes\sigma$ is not semisimple as well?
I'm expecting the resulting representation to be not semisimple, but I need an answer to my question to be able to prove it.

Consider two (possibly infinite-dimensional) representations $\rho$, $\pi$ of a semisimple Lie algebra $\mathfrak{g}$, with $\rho$ irreducible and $\pi$ indecomposable but not irreducible (i.e., not simple). Are there any circumstances under which the product representation $\rho\otimes\pi$ can be semisimple (isomorphic to a direct sum of irreducible representations)?

In other words: is the product of a simple and a non-semisimple representation always non-semisimple?

Motivation

I'm working in particular with $\mathfrak{spin}(3,1)$. The situation is the following: I have a class of irreducible finite-dimensional representations (the $\rho$ above) that I know how to couple with generic irreducible representations $\sigma$ (that is, I know under which conditions the resulting representation is semisimple or not).
I need to find out if the product $\rho_1\otimes\rho_2\otimes\sigma$ is semisimple: if $\rho_2\otimes\sigma$ is, then I can consider the product with $\rho_1$ for each one of the irreducibles in the decomposition to obtain the answer; if, however, $\rho_2\otimes\sigma$ is indecomposable but not irreducible, can I be sure that $\rho_1\otimes\rho_2\otimes\sigma$ is not semisimple as well?
I'm expecting the resulting representation to be not semisimple, but I need an answer to my question to be able to prove it.

Update #1

As Will Sawin pointed out, if $\rho$ is finite dimensional and $\rho\otimes\pi$ happens to be semisimple, then it must necessarily be $\rho^*\otimes\rho\otimes\pi$ non-semisimple, since $\rho^*\otimes\rho$ is semisimple and contains the trivial representation in its decomposition. Unfortunately this doesn't contradict anything, as it is possible for the product of two semisimple representations to be non-semisimple.

Consider two (possibly infinite-dimensional) representations $\rho$, $\pi$ of a semisimple Lie algebra $\mathfrak{g}$, with $\rho$ irreducible and $\pi$ indecomposable but not irreducible (i.e., not simple). Are there any circumstances under which the product representation $\rho\otimes\pi$ can be semisimple (isomorphic to a direct sum of irreducible representations)?

In other words: is the product of a semisimple and a non-semisimple representation always non-semisimple?

Motivation

I'm working in particular with $\mathfrak{spin}(3,1)$. The situation is the following: I have a class of irreducible finite-dimensional representations (the $\rho$ above) that I know how to couple with generic irreducible representations $\sigma$ (that is, I know under which conditions the resulting representation is semisimple or not).
I need to find out if the product $\rho_1\otimes\rho_2\otimes\sigma$ is semisimple: if $\rho_2\otimes\sigma$ is, then I can consider the product with $\rho_1$ for each one of the irreducibles in the decomposition to obtain the answer; if, however, $\rho_2\otimes\sigma$ is indecomposable but not irreducible, can I be sure that $\rho_1\otimes\rho_2\otimes\sigma$ is not semisimple as well?
I'm expecting the resulting representation to be not semisimple, but I need an answer to my question to be able to prove it.

Consider two (possibly infinite-dimensional) representations $\rho$, $\pi$ of a semisimple Lie algebra $\mathfrak{g}$, with $\rho$ irreducible and $\pi$ indecomposable but not irreducible (i.e., not simple). Are there any circumstances under which the product representation $\rho\otimes\pi$ can be semisimple (isomorphic to a direct sum of irreducible representations)?

In other words: is the product of a simple and a non-semisimple representation always non-semisimple?

Motivation

I'm working in particular with $\mathfrak{spin}(3,1)$. The situation is the following: I have a class of irreducible finite-dimensional representations (the $\rho$ above) that I know how to couple with generic irreducible representations $\sigma$ (that is, I know under which conditions the resulting representation is semisimple or not).
I need to find out if the product $\rho_1\otimes\rho_2\otimes\sigma$ is semisimple: if $\rho_2\otimes\sigma$ is, then I can consider the product with $\rho_1$ for each one of the irreducibles in the decomposition to obtain the answer; if, however, $\rho_2\otimes\sigma$ is indecomposable but not irreducible, can I be sure that $\rho_1\otimes\rho_2\otimes\sigma$ is not semisimple as well?
I'm expecting the resulting representation to be not semisimple, but I need an answer to my question to be able to prove it.

Consider two (possibly infinite-dimensional) representations $\rho$, $\pi$ of a semisimple Lie algebra $\mathfrak{g}$, with $\rho$ irreducible and $\pi$ indecomposable but not irreducible (i.e., not simple). Are there any circumstances under which the product representation $\rho\otimes\pi$ can be semisimple (isomorphic to a direct sum of irreducible representations)?

Motivation

I'm working in particular with $\mathfrak{spin}(3,1)$. The situation is the following: I have a class of irreducible finite-dimensional representations (the $\rho$ above) that I know how to couple with generic irreducible representations $\sigma$ (that is, I know under which conditions the resulting representation is semisimple or not).
I need to find out if the product $\rho_1\otimes\rho_2\otimes\sigma$ is semisimple: if $\rho_2\otimes\sigma$ is, then I can consider the product with $\rho_1$ for each one of the irreducibles in the decomposition to obtain the answer; if, however, $\rho_2\otimes\sigma$ is indecomposable but not irreducible, can I be sure that $\rho_1\otimes\rho_2\otimes\sigma$ is not semisimple as well?
I'm expecting the resulting representation to be not semisimple, but I need an answer to my question to be able to prove it.

Consider two (possibly infinite-dimensional) representations $\rho$, $\pi$ of a semisimple Lie algebra $\mathfrak{g}$, with $\rho$ irreducible and $\pi$ indecomposable but not irreducible (i.e., not simple). Are there any circumstances under which the product representation $\rho\otimes\pi$ can be semisimple (isomorphic to a direct sum of irreducible representations)?

In other words: is the product of a semisimple and a non-semisimple representation always non-semisimple?

Motivation

I'm working in particular with $\mathfrak{spin}(3,1)$. The situation is the following: I have a class of irreducible finite-dimensional representations (the $\rho$ above) that I know how to couple with generic irreducible representations $\sigma$ (that is, I know under which conditions the resulting representation is semisimple or not).
I need to find out if the product $\rho_1\otimes\rho_2\otimes\sigma$ is semisimple: if $\rho_2\otimes\sigma$ is, then I can consider the product with $\rho_1$ for each one of the irreducibles in the decomposition to obtain the answer; if, however, $\rho_2\otimes\sigma$ is indecomposable but not irreducible, can I be sure that $\rho_1\otimes\rho_2\otimes\sigma$ is not semisimple as well?
I'm expecting the resulting representation to be not semisimple, but I need an answer to my question to be able to prove it.