This may be helpful.

In the below paper it is proved that every element in the full matrix algebra $M_3(k)$ is a sum of two idempotents if and only if every polynomial of degree $3$ has a root in $k$.

de Seguins Pazzis, Clément; On decomposing any matrix as a linear combination of three idempotents. Linear Algebra Appl. 433 (2010), no. 4, 843–855.

This may be helpful.

In the below paper it is proved that every element in the full matrix algebra $M_3(k)$ is a sum of two idempotents if and only if every polynomial of degree $3$ has a root in $k$.

de Seguins Pazzis, Clément; On decomposing any matrix as a linear combination of three idempotents. Linear Algebra Appl. 433 (2010), no. 4, 843–855.

Added: the name "cubically closed field" was almost suggested by a removed comment of @EricWofsey. And it seems that it is the right name for these fields.

A partial answer:

If $k$ is an ordered (formally real) field then it should be a real closed field. Since every irreducible polynomial of degree $3$ gives rise a formally real extension of degree $3$. So $k$ is elementary equivalent to real numbers.

In the below paper it is proved that every element in the full matrix algebra $M_3(k)$ is a sum of two idempotents if and only if every polynomial of degree $3$ has a root in $k$.

de Seguins Pazzis, Clément; On decomposing any matrix as a linear combination of three idempotents. Linear Algebra Appl. 433 (2010), no. 4, 843–855.

This may be helpful.

In the below paper it is proved that every element in the full matrix algebra $M_3(k)$ is a sum of two idempotents if and only if every polynomial of degree $3$ has a root in $k$.

de Seguins Pazzis, Clément; On decomposing any matrix as a linear combination of three idempotents. Linear Algebra Appl. 433 (2010), no. 4, 843–855.