On the $\mathbb{A}^1$-invariance of $\mathrm{K}_2$ modeled on linear and even orthogonal groups

. Let k be an arbitrary field. In this paper we show that in the linear case ( Φ = A (cid:96) , (cid:96) ≥ 4 ) and even orthogonal case ( Φ = D (cid:96) , (cid:96) ≥ 7 , char( k ) (cid:54) = 2 ) the unstable functor K 2 (Φ , − ) possesses the A 1 invariance property in the geometric case, i. e. K 2 (Φ , R [ t ]) = K 2 (Φ , R ) for a regular ring R containing k . As a consequence, the unstable K 2 groups can be represented in the unstable A 1 -homotopy category H • ( k ) as fundamental groups of the simply-connected Chevalley–Demazure group schemes G(Φ , − ) . Our invariance result can be considered as the K 2 -analogue of the geometric case of Bass–Quillen conjecture. We also show for a semilocal regular k -algebra A that K 2 (Φ , A ) embeds as a subgroup into K M2 (Frac A ) .