A proof of Frobenius Theorem on division algebras and related results

There will be considered a short proof of the Frobenius classification of finite-dimensional real division algebras. Then we will consider results relating the presence in an algebra a subalgebra isomorphic to the algebra of complex numbers or to the quaternion algebra with the structure of its ideals. Then we discuss general conditions providing lifting of algebraic elements from quotients of an algebra (not necessarily real or finite-dimensional) by arbitrary ideals or nil-ideals.
This is a joint work with Matej Bresar.