For every n, we construct a sum-of-squares identity
(\sum_{i=1}^n x_i^2) (\sum_{j=1}^n y_j^2)= \sum_{k=1}^s f_k^2,
where f_k are bilinear forms with complex coefficients and s= O(n^{1.62}). Previously, such a construction was known with s=O(n^2/\log n).
The same bound holds over any field of positive characteristic.

As an application to complexity of non-commutative computation, we show that the polynomial ID_n=\sum_{i,j\in [n]}x_iy_jx_iy_j in 2n non-commuting variables can be computed by a non-commutative arithmetic circuit of size O(n^{1.96}). This holds over any field of characteristic different from two. The same bound applies to non-commutative versions of the elementary symmetric polynomial of degree four and the rectangular permanent of a 4\times n matrix.