If you want to learn more about this subject, I'd suggest looking at "A survey of the S-Lemma" by Imre Polik and Tamas Terlaky, from SIAM Review, Vol 49, No. 3, p 371-418, 2007. In particular, look at section 6 of the paper that discusses problems with rank limitations (you want your B matrix to be rank 1.) After reviewing this paper and thinking about your problem a bit, I can't see how to apply the S-Lemma approach to your particular problem.

CSDP (and all of the other SDP solvers) support block diagonal matrices and also allow for blocks which are stricted to being block diagonal. If $B$ is of size $N$ by $N$, then your X matrix will have one block of size $N+1$ by $N+1$, holding $B$, $\beta$, and the $1$. You'll need one equality constraint to specify that this first element is 1. Your X matrix will also have a diagonal block of size $N$ to hold the slack variables. There's no need for constraints to make those off diagonal elements 0. If you're using the MATLAB interface to CSDP, then you'd have K.s=N+1 and K.l=N.

The general result is that if you have a quadratically constrained quadratic programming problem with only one non-convex constraint, then you can formulate it exactly as an SDP. That's why I believe that this relaxation will be exact in this case. However, I haven't looked into this for several years, so the details escape me at the moment. If I get a chance I'll try to sit down and work through the details.

See appendix B of "Convex Optimization" by Boyd and Vandenberghe.

On second thought, although approach I sketched out is sufficient to show that the SDP is a relaxation of the original problem, it's not enough to show that the relaxation is exact (I forgot to account for the $L^{T}CL$ term in the constraint. I'm afraid that you need the machinery of the S-lemma to show that the relaxation is exact.

As I recall, you can also get to this formulation using the S-lemma.

There are several software packages for solving such an SDP, including SDPA (written in C++) and CSDP (in C)

Your notation seems a bit confused (and perhaps suggests why you're unable to establish this result.) You haven't explained what $a_{0}$, $a_{1}$, $...$ are.

The poster has also put this same question up on scicomp.stackexchange.com, where she's somewhat more likely to get a useful answer, provided that she can clarify the question.