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JavaScript

// split string in brackets

var str1 = " {{a}} {{b}} aaa", str2 = "{{a {{b}}}}", str3 = "{{{a}}};"

var splits = str1.split(/[^{][^.][^}]|[{}]/).filter(Boolean);
console.log(splits);

var splits = str2.split(/[^{][^.][^}]|[{}]/).filter(Boolean);
console.log(splits);

var splits = str3.split(/[^{][^.][^}]|[{}]/).filter(Boolean);
console.log(splits);