For a root system Φ of type E ℓ and arbitrary commutative ring R we show that the group K 2 (Φ, R) is contained in the center of the Steinberg group St(Φ, R). In course of the proof we also demonstrate an analogue of Quillen-Suslin local-global principle for K 2 (Φ, R).It is classically known that K 2 (Φ, R) is contained in the center of St(Φ, R) if R is a local ring and Φ has rank ≥ 2 (see [14, Theorem 2.13]). One of the standard conjectures in the theory of Chevalley groups over rings asserts that St(Φ, R) is a central extension of E(Φ, R) for any commutative ring R provided that the rank of Φ is sufficiently large. We refer to this conjecture as centrality of K 2 . It can be regarded as a "K 2 -analogue" of Taddei's normality theorem. For Chevalley groups of rank 2 centrality of K 2 fails already for 1-dimensional R (see [25, Theorem 1]).W. van der Kallen in [5] and recently A. Lavrenov in [7] proved centrality of K 2 for Φ = A ℓ , C ℓ , ℓ ≥ 3. The main technical ingredient of both proofs is the so-called method of "another presentation" which consists in presenting St(Φ, R) as a group with a set of generators modeling elements of root type (see [22, § 4] for the idea of the definition). For example, in the linear case the elements of root type are exactly the usual linear transvections, see Section 2.3. The key advantage of this method lies in the fact that it allows one to define an action of G(Φ, R) on St(Φ, R) which turns the canonical map φ into a crossed module.Recall that a crossed module is a triple C = (N, M, µ) consisting of an abstract group N acting on itself by conjugation, an N-group M and a group morphism µ : M → N which preserves the action of N and satisfies Peiffer identity µ(m) · m ′ = mm ′ m −1 , m, m ′ ∈ M. Date: October 19, 2018.

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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.

We prove the injective stability theorem for unitary K 1 under the usual stable range condition on the ground ring. This improves the stability theorem of A. Bak, V. Petrov and G. Tang where a stronger ƒ-stable range condition was used.