You are actually looking to solve the continuous algebraic Riccati equation. For convenience, I will write your $B$ as $X=X^T$. Then the equation you're trying to solve is simply

$$ X - XAX + (-C) = 0$$

Or even more explicitly, writing the Cholesky factorization of $A=BB^T$

$$ \left( \frac{1}{2}I \right)^TX + X\left( \frac{1}{2}I \right) - XBB^TX + (-C) = 0$$

An algebraic riccati equation can be efficiently solved using Hamiltonian matrices. But I would instead refer you to the "care" command in MATLAB.

You are actually looking to solve the continuous algebraic Riccati equation. For convenience, I will write your $B$ as $X=X^T$. Then the equation you're trying to solve is simply

$$ X - XAX + (-C) = 0$$

Or even more explicitly, writing the Cholesky factorization of $A=BB^T$

$$ \left( \frac{1}{2}I \right)^TX + X\left( \frac{1}{2}I \right) - XBB^TX + (-C) = 0$$

The solution of an algebraic Riccati equation using Hamiltonian matrices is a standard topic in control theory. But I would instead refer you to the "care" command in MATLAB.