  We obtain new proofs with improved constants of the Khintchine-type
inequality with matrix coefficients in two cases. The first case is the Pisier
and Lust-Piquard noncommutative Khintchine inequality for $p=1$, where we
obtain the sharp lower bound of $\frac1{\sqrt{2}}$ in the complex Gaussian case
and for the sequence of functions $\{e^{i2^nt}\}_{n=1}^\infty$ . The second
case is Junge's recent Khintchine-type inequality for subspaces of the operator
space $R\oplus C$, which he used to construct a cb-embedding of the operator
Hilbert space $OH$ into the predual of a hyperfinite factor. Also in this case,
we obtain a sharp lower bound of $\frac1{\sqrt{2}}$ . As a consequence, it
follows that any subspace of a quotient of $(R\oplus C)^*$ is cb-isomorphic to
a subspace of the predual of the hyperfinite factor of type $III_1$, with
cb-isomorphism constant $\leq \sqrt{2}$ . In particular, the operator Hilbert
space $OH$ has this property.
