Assume that I have a finitely presented algebra $A$ over the complex numbers (by which I mean that $A$ is generated over $\mathbb{C}$ by finitely many elements $x_1,...,x_n$ subject to finitely many relations). Let now $y_1,\ldots, y_m$ be a finite subset of elements of $A$. Is there an algorithm which gives a presentation of the subalgebra $B$ of $A$ generated by $y_1,...y_m$ ? In particular, I am interested in the following example: Let $A = \mathbb{C}B_3 = \mathbb{C}\langle \sigma_1,\sigma_2 | \sigma_1\sigma_2\sigma_1 = \sigma_2\sigma_1\sigma_2\rangle$ be the group algebra of the Braid group on 3 braids. Is there any nice presentation of the subalgebra generated by $\sigma_1$ and $\sigma_2 + \sigma_2^{-1}$ ?