The Grothendieck inequality states that there is a universal constant ${\displaystyle K_{G}}$ with the following property. If $M_{i,j}$ is an $n$ by $n$ (real or complex) matrix with $$\left|\sum_{i,j}^{ }M_{i,j}s_{i}t_{j}\right|\le1$$ for all (real or complex) numbers $s_i, t_j$ of absolute value at most $1$, then $$\left|\sum_{i,j}^{ }M_{i,j}\left\langle S_{i},T_{j}\right\rangle \right|\le K_{G}$$ for all vectors $S_i, T_j$ in the unit ball $B(H)$ of a (real or complex) Hilbert space $H$.
It's known that the sequences $K_{G}^{\mathbb R}(d)$ and $K_{G}^{\mathbb C}(d)$ are increasing,moreover the bounds are given by:
$$\frac{\pi}{2}\le K_{G}^{\mathbb R}(d)\le\sinh\left(\frac{\pi}{2}\right)$$
The proof of this claim is from a paper of Grothendieck,unfortunately I have not found it.
Can someone prove the inequality using the same approach as Grothendieck's?