We solve a classical problem of centrality of symplectic K 2 , namely we show that for an arbitrary commutative ring R, l ≥ 3, the symplectic Steinberg group StSp(2l, R) as an extension of the elementary symplectic group Ep(2l, R) is a central extension. This allows to conclude that the explicit definition of symplectic K 2 Sp(2l, R) as a kernel of the above extension, i.e. as a group of non-elementary relations among symplectic transvections, coincides with the usual implicit definition via plus-construction.We proceed from van der Kallen's classical paper, where he shows an analogous result for linear K-theory. We find a new set of generators for the symplectic Steinberg group and a defining system of relations among them. In this new presentation it is obvious that the symplectic Steinberg group is a central extension.

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We prove the centrality of K2(F4, R) for an arbitrary commutative ring R. This completes the proof of the centrality of K2(Φ, R) for any root system Φ of rank ≥ 3. Our proof uses only elementary localization techniques reformulated in terms of pro-groups. Another new result of the paper is the construction of a crossed module on the canonical homomorphism St(Φ, R) → Gsc(Φ, R), which has not been known previouly for exceptional Φ.

Let (R, Λ) be an arbitrary form ring, let U (2n, R, Λ) denote the hyperbolic unitary group, let EU(2n, R, Λ) be its elementary subgroup and StU(2n, R, Λ) the unitary Steinberg group. It is proved that, if n ≥ 5 (a natural assumption for similar results), then every central extension of StU(2n, R, Λ) splits. This results makes it possible to describe the Schur multiplier of the elementary unitary group as the kernel of the natural epimorphism of StU(2n, R, Λ) onto EU(2n, R, Λ) if it is known that this kernel is included in the center of the unitary Steinberg group. Steinberg's description of relations is employed, which leads to simplest proofs of these results.

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