We prove that, if $M > 4(1+2\sqrt{3})$ and $\varepsilon > 0$, if $\mathcal{V}$ and $\mathcal{W}$ are complex JBW*-triples (with preduals $\mathcal{V}_{*}$ and $\mathcal{W}_{*}$, respectively), and if $U$ is a separately weak*-continuous bilinear form on $\mathcal{V} \times \mathcal{W}$, then there exist norm-one functionals $\varphi_{1},\varphi_{2}\in \mathcal{V}_{*}$ and $\psi_{1},\psi_{2}\in \mathcal{W}_{*}$ satisfying
|U(x,y)| \leq M \,\|U\| ( \|x\|_{\varphi_{2}}^{2} + \varepsilon^{2} \, \|x\|_{\varphi_{1}}^{2} )^{\frac 12} ( \|y\|_{\psi_{2}}^{2} + \varepsilon^{2} \, \|y\|_{\psi_{1}}^{2} )^{\frac 12}
for all $(x,y)\in \mathcal{V} \times \mathcal{W}$. Here, for a norm-one functional $\varphi$ on a complex JB*-triple $\mathcal{V}$, $\|\cdot\|_{\varphi}$ stands for the prehilbertian seminorm on $\mathcal{V}$ associated to $\varphi$ given by $\|x\|_{\varphi}^{2} := \varphi \{x,x,z\}$ for all $x\in \mathcal{W}$, where $z\in \mathcal{V}^{**}$ satisfies $\varphi (z) = \|z\|=1$. We arrive at this form of ‘Grothendieck's inequality’ through results of C.-H. Chu, B. Iochum, and G. Loupias, and an amended version of the ‘little Grothendieck's inequality’ for complex JB*-triples due to T. Barton and Y. Friedman. We also obtain extensions of these results to the setting of real JB*-triples. 2000 Mathematical Subject Classification: 17C65, 46K70, 46L05, 46L10, 46L70.