For 1: a field extension $L/k$ is separable is $L$ is a separable algebra, that is, if $L\otimes K$ is a semisimple algebra for all field extensions $K/k$. In particular, if $L$ is separable over $k$ and $K$ is an extension of $k$, then $L\otimes K$ will have no nilpotent elements because it is semisimple.

For details, see for example, Pierce's beautiful book Associative algebras.

For (a): a field extension $L/k$ is separable is $L$ is a separable algebra, that is, if $L\otimes K$ is a semisimple algebra for all field extensions $K/k$. In particular, if $L$ is separable over $k$ and $K$ is an extension of $k$, then $L\otimes K$ will have no nilpotent elements because it is semisimple.

For details, see for example, Pierce's beautiful book Associative algebras.

For (b), for which the argument is more elaborate, see Jacobson's Lectures in Abstract Algebra, vol. 3, where the result is part 3 of Theorem 21; he does without algebraicity, by the way.