Given a basis $U$, what conditions are needed for an orthogonal basis for it?

Question

Given a basis $U$, what conditions are needed for an orthogonal basis for it?

For example, in the following vector space $U$, if $U =sp\{(1,1,1),(1,3,7)\}$ then what conditions are needed for an orthogonal basis for it?

Is it enough to have a basis of dimension $2$ that's orthogonal? or are there more conditions?

EDIT: If for example I find an orthogonal span of dimension 2, say $V=sp\{(1,1,1), v_2\} $ such that $v_2$ is orthogonal to $(1,1,1)$, is any vector that's orthogonal to $(1,1,1)$ fine for it to be an orthogonal span for $U$?

PS: I know there's GS algorithm, but I'm asking if other bases that we get in other ways are also fine.

Answer 1

Since $U=sp\{u_1,u_2\}$ is a plane, then any span where $sp\{u_1,v_1\}: v_1\perp u_1$ doesn't necessarily span the same plane as U. So we we can't just take any vector that's orthogonal to $u_1$ as $v_1$.

Thanks Gerry Myerson.