Let's say that we have a complex isometry matrix $M$ of size $m \times n$, with $m \geq n$ and $M^\dagger M = \mathbb{1}$. Is it possible to enlarge such a matrix, e.g., to a size $m \times (n + k)$, as long as $m \geq n + k$, while preserving its initial entries, so that it is still an isometry?
Increasing the $m$ dimension would also be interesting, but it's not crucial.
At the end I saw you work in complex setting. Please replace Euclidean and $\mathbb R^j$ all with $\mathbb C^j$.
Enlarging to $(m+k) \times (n+k)$ should be trivial: put $k$ many $1$ onto the shifted diagonal (shifted by $m-n$). Other additional entries are zero.
Enlarging to $m\times m$ should not be difficult: $M$ represents an isometry $M\colon X\to Y$, of Euclidean spaces. Let $Z$ be the orthogonal complement of $M(X)$ in $Y$, let $v_{n+1},\dots,v_m$ be an orthonormal base of $Z$.
Let also $e_1,\dots,e_m$ be the canonical base of $\mathbb R^m \supset \mathbb R^n = X$.
Extend $M$ linearly by sending $e_{n+1},\dots,e_m$ to $v_{n+1},\dots,v_m$.
Does this solve your problem?