Let \(C\) be a projective complex algebraic curve with two distinguished points \(P_{\pm}\) and let \(A\) be the ring of meromorphic functions on \(C\), holomorphic outside \(\{P_{\pm}\}\). Let \(\mathfrak g\) be a complex finite dimensional simple Lie algebra. The Lie algebra \(\mathfrak g\otimes A\) and a one-dimensional central extension \(\mathfrak g{\hat \otimes} A\) were first studied in [\textit{I. M. Krichever} and \textit{S. P. Novikov}, Funkts. Anal. Prilozh. 21, No. 2, 46--63 (1987; Zbl 0634.17010); see also ibid. 21, No. 4, 47--61 (1988; Zbl 0659.17012)]. (If the genus of \(C\) is \(0\), \(\mathfrak g{\hat \otimes} A\) is an affine Kac-Moody algebra). As in the Kac-Moody case, \(\mathfrak g{\hat \otimes} A\) has also further extensions \(\mathfrak g{\tilde \otimes} A=\mathfrak g {\hat \otimes} A\oplus \mathbb C e\), where \(e\) is a meromorphic vector field on \(C\), holomorphic outside \(\{P_{\pm}\}\). The paper under review is concerned with the case when the genus of \(C\) is 1, i.e. when \(C\) is elliptic. The author exhibits a set of generators of \(\mathfrak g{\tilde \otimes} A\) and the main relations satisfied by them; the problem of finding a presentation remains open. Let \(m\) be the number of zeros of \(e\) outside \(\{P_{\pm}\}\). It is shown that \(\mathfrak g{\tilde \otimes} A\) admits \(m+1\) linearly independent invariant symmetric bilinear forms. When \(m=0\), the author proves that \(\mathfrak g{\tilde \otimes} A\) has a unique such form. Next, a correspondence between the algebras \(\mathfrak g\otimes A\) and the complex crystallographic Coxeter groups [cf. \textit{I. N. Bernshteĭn} and \textit{O. V. Shvartsman}, Funkts. Anal. Prilozh. 12, No. 4, 79--80 (1977; Zbl 0458.32017)] is established. Finally, the author studies the orbits on \(\mathfrak g{\tilde \otimes} A\) under the adjoint action of the group corresponding to \(\mathfrak g\otimes A\).