Amicable Hadamard pairs in every order divisible by four

Construct an amicable pair of Hadamard matrices for every order divisible by 4.

Background

An amicable pair consists of a symmetric Hadamard matrix and a skew Hadamard matrix satisfying the requisite commutation relation. Such pairs are central to one of the paper’s ETF multiplication constructions. The paper states that their existence in every order divisible by 4 is conjectural.

References

The notion of amicable pairs appeared in and, conjecturally, amicable pairs also exist for all $N$ divisible by 4 .

— New constructions of optimal arrangements of $2d$ lines in $\mathbb{C}^d$  (2608.16116 - Glazyrin, 17 Aug 2026) in Section 2, Hadamard and conference matrices